HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 15.9s

Alternatives: 14

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(fma(Float32(Float32(1.0) - 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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 5: 91.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-0.5 \cdot \frac{\frac{\left(\left(u \cdot 16 - u \cdot 8\right) + 9.333333333333334 \cdot \frac{u}{v}\right) - \left(4 \cdot \frac{u \cdot 8 - u \cdot 16}{v} + \frac{u}{v} \cdot 32\right)}{v} - u \cdot -4}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (*
       -0.5
       (/
        (-
         (/
          (-
           (+ (- (* u 16.0) (* u 8.0)) (* 9.333333333333334 (/ u v)))
           (+ (* 4.0 (/ (- (* u 8.0) (* u 16.0)) v)) (* (/ u v) 32.0)))
          v)
         (* u -4.0))
        v))
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((-0.5f * (((((((u * 16.0f) - (u * 8.0f)) + (9.333333333333334f * (u / v))) - ((4.0f * (((u * 8.0f) - (u * 16.0f)) / v)) + ((u / v) * 32.0f))) / v) - (u * -4.0f)) / v)) + (v * (-1.0f + (1.0f / expf((-2.0f / v)))))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (((-0.5e0) * (((((((u * 16.0e0) - (u * 8.0e0)) + (9.333333333333334e0 * (u / v))) - ((4.0e0 * (((u * 8.0e0) - (u * 16.0e0)) / v)) + ((u / v) * 32.0e0))) / v) - (u * (-4.0e0))) / v)) + (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v)))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))) + Float32(Float32(9.333333333333334) * Float32(u / v))) - Float32(Float32(Float32(4.0) * Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) / v)) + Float32(Float32(u / v) * Float32(32.0)))) / v) - Float32(u * Float32(-4.0))) / v)) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(-0.5) * (((((((u * single(16.0)) - (u * single(8.0))) + (single(9.333333333333334) * (u / v))) - ((single(4.0) * (((u * single(8.0)) - (u * single(16.0))) / v)) + ((u / v) * single(32.0)))) / v) - (u * single(-4.0))) / v)) + (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v)))))));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

   3. Taylor expanded in u around 0 68.5%

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

   4. Taylor expanded in v around -inf 68.3%

      \[\leadsto u \cdot \left(-0.5 \cdot \color{blue}{\left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{\left(-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + 9.333333333333334 \cdot \frac{u}{v}\right) - \left(4 \cdot \frac{8 \cdot u - 16 \cdot u}{v} + 32 \cdot \frac{u}{v}\right)}{v}}{v}\right)} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-0.5 \cdot \frac{\frac{\left(\left(u \cdot 16 - u \cdot 8\right) + 9.333333333333334 \cdot \frac{u}{v}\right) - \left(4 \cdot \frac{u \cdot 8 - u \cdot 16}{v} + \frac{u}{v} \cdot 32\right)}{v} - u \cdot -4}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v} + -0.5 \cdot \left(4 \cdot \left(u + -1\right) + -4 \cdot \left({u}^{2} \cdot \left(-1 + \frac{2 + \frac{-1}{u}}{u}\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (+
       (* 0.16666666666666666 (/ (* u (- 8.0 (* u (+ 24.0 (* u -16.0))))) v))
       (*
        -0.5
        (+
         (* 4.0 (+ u -1.0))
         (* -4.0 (* (pow u 2.0) (+ -1.0 (/ (+ 2.0 (/ -1.0 u)) u)))))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((0.16666666666666666f * ((u * (8.0f - (u * (24.0f + (u * -16.0f))))) / v)) + (-0.5f * ((4.0f * (u + -1.0f)) + (-4.0f * (powf(u, 2.0f) * (-1.0f + ((2.0f + (-1.0f / u)) / u))))))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (((0.16666666666666666e0 * ((u * (8.0e0 - (u * (24.0e0 + (u * (-16.0e0)))))) / v)) + ((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) + ((-4.0e0) * ((u ** 2.0e0) * ((-1.0e0) + ((2.0e0 + ((-1.0e0) / u)) / u))))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(8.0) - Float32(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))))) / v)) + Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) + Float32(Float32(-4.0) * Float32((u ^ Float32(2.0)) * Float32(Float32(-1.0) + Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) / u)) / u))))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(0.16666666666666666) * ((u * (single(8.0) - (u * (single(24.0) + (u * single(-16.0)))))) / v)) + (single(-0.5) * ((single(4.0) * (u + single(-1.0))) + (single(-4.0) * ((u ^ single(2.0)) * (single(-1.0) + ((single(2.0) + (single(-1.0) / u)) / u))))))) / v));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

      1. +-commutative 95.0%

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

      2. fma-define 95.1%

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

      3. +-commutative 95.1%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

      1. fma-undefine 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in v around -inf 68.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} + 1 \]

   8. Taylor expanded in u around 0 68.1%

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

   9. Taylor expanded in u around -inf 68.1%

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

       1. mul-1-neg 68.1%

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

       2. unsub-neg 68.1%

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

   11. Simplified 68.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v} + -0.5 \cdot \left(4 \cdot \left(u + -1\right) + -4 \cdot \left({u}^{2} \cdot \left(-1 + \frac{2 + \frac{-1}{u}}{u}\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot \left(1 + u \cdot \left(u - 2\right)\right)\right) - 0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (+ 1.0 (* u (- u 2.0))))))
       (* 0.16666666666666666 (/ (* u (- (* u (+ 24.0 (* u -16.0))) 8.0)) v)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * (1.0f + (u * (u - 2.0f)))))) - (0.16666666666666666f * ((u * ((u * (24.0f + (u * -16.0f))) - 8.0f)) / v))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * (1.0e0 + (u * (u - 2.0e0)))))) - (0.16666666666666666e0 * ((u * ((u * (24.0e0 + (u * (-16.0e0)))) - 8.0e0)) / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * Float32(Float32(1.0) + Float32(u * Float32(u - Float32(2.0))))))) - Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))) - Float32(8.0))) / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * (single(1.0) + (u * (u - single(2.0))))))) - (single(0.16666666666666666) * ((u * ((u * (single(24.0) + (u * single(-16.0)))) - single(8.0))) / v))) / v));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

      1. +-commutative 95.0%

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

      2. fma-define 95.1%

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

      3. +-commutative 95.1%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

      1. fma-undefine 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in v around -inf 68.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} + 1 \]

   8. Taylor expanded in u around 0 68.1%

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

   9. Taylor expanded in u around 0 68.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot \left(1 + u \cdot \left(u - 2\right)\right)\right) - 0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v} - -0.5 \cdot \left(u \cdot \left(u \cdot -4 + 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* 0.16666666666666666 (/ (* u (- 8.0 (* u (+ 24.0 (* u -16.0))))) v))
       (* -0.5 (* u (+ (* u -4.0) 4.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((0.16666666666666666f * ((u * (8.0f - (u * (24.0f + (u * -16.0f))))) / v)) - (-0.5f * (u * ((u * -4.0f) + 4.0f)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (((0.16666666666666666e0 * ((u * (8.0e0 - (u * (24.0e0 + (u * (-16.0e0)))))) / v)) - ((-0.5e0) * (u * ((u * (-4.0e0)) + 4.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(8.0) - Float32(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))))) / v)) - Float32(Float32(-0.5) * Float32(u * Float32(Float32(u * Float32(-4.0)) + Float32(4.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(0.16666666666666666) * ((u * (single(8.0) - (u * (single(24.0) + (u * single(-16.0)))))) / v)) - (single(-0.5) * (u * ((u * single(-4.0)) + single(4.0))))) / v));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

      1. +-commutative 95.0%

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

      2. fma-define 95.1%

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

      3. +-commutative 95.1%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

      1. fma-undefine 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in v around -inf 68.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} + 1 \]

   8. Taylor expanded in u around 0 68.1%

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

   9. Taylor expanded in u around 0 68.1%

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

       1. *-commutative 68.1%

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

   11. Simplified 68.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v} - -0.5 \cdot \left(u \cdot \left(u \cdot -4 + 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/ (+ 1.3333333333333333 (* (- (* u 8.0) (* u 16.0)) 0.5)) v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f + (((u * 8.0f) - (u * 16.0f)) * 0.5f)) / v) - (u * 2.0f))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + (((u * 8.0e0) - (u * 16.0e0)) * 0.5e0)) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5))) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + (((u * single(8.0)) - (u * single(16.0))) * single(0.5))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

   3. Taylor expanded in u around 0 68.5%

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

   4. Taylor expanded in v around -inf 60.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (* u (- (+ 2.0 (/ 1.3333333333333333 v)) (* u (+ 2.0 (/ 4.0 v)))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * ((2.0f + (1.3333333333333333f / v)) - (u * (2.0f + (4.0f / v))))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * ((2.0e0 + (1.3333333333333333e0 / v)) - (u * (2.0e0 + (4.0e0 / v))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) - Float32(u * Float32(Float32(2.0) + Float32(Float32(4.0) / v))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * ((single(2.0) + (single(1.3333333333333333) / v)) - (u * (single(2.0) + (single(4.0) / v))))) / v));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

      1. +-commutative 95.0%

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

      2. fma-define 95.1%

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

      3. +-commutative 95.1%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

      1. fma-undefine 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in v around -inf 68.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} + 1 \]

   8. Taylor expanded in u around 0 68.1%

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

   9. Taylor expanded in u around 0 60.6%

      \[\leadsto \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\color{blue}{u \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 + 1.3333333333333333 \cdot \frac{1}{v}\right)\right)}}{v}\right) + 1 \]
   10. Step-by-step derivation

       1. associate-*r/ 60.6%

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

       2. metadata-eval 60.6%

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

       3. associate-*r/ 60.6%

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

       4. metadata-eval 60.6%

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

   11. Simplified 60.6%

       \[\leadsto \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\color{blue}{u \cdot \left(u \cdot \left(2 + \frac{4}{v}\right) - \left(2 + \frac{1.3333333333333333}{v}\right)\right)}}{v}\right) + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - u \cdot 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (+ 2.0 (/ (- 2.0 (* u 2.0)) v))))))

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 - (u * 2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) - (u * single(2.0))) / v)));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

   3. Taylor expanded in u around 0 68.5%

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

   4. Taylor expanded in v around -inf 56.5%

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

4. Final simplification 91.1%

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

Alternative 12: 90.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = 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

   3. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 95.0%

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

   3. Taylor expanded in u around 0 57.4%

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

   4. Taylor expanded in v around inf 52.8%

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

      1. sub-neg 52.8%

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

      2. distribute-lft-out 52.8%

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

      3. metadata-eval 52.8%

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

   6. Simplified 52.8%

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

4. Final simplification 90.9%

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

Alternative 13: 87.0% accurate, 213.0× 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 87.6%

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

Alternative 14: 5.8% accurate, 213.0× 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 5.3%

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

Reproduce

herbie shell --seed 2024090 
(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))))))))