HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.1s

Alternatives: 15

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. expm1-log1p-u 90.9%

      \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

   2. log1p-define 91.0%

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

   3. expm1-undefine 91.0%

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

   4. add-exp-log 99.3%

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

   5. +-commutative 99.3%

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

   6. +-commutative 99.3%

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

   7. fma-undefine 99.4%

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

   8. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 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 99.3%

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

3. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. fma-define 99.4%

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

5. Simplified 99.4%

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

Alternative 3: 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 99.3%

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

Alternative 4: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in u around 0 94.6%

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

Alternative 5: 91.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - \left(u \cdot \left(8 + u \cdot -4\right) - 4\right)\right) - 0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* -0.5 (- (* 4.0 (+ u -1.0)) (- (* u (+ 8.0 (* u -4.0))) 4.0)))
       (* 0.16666666666666666 (/ (* u (- (* u (+ 24.0 (* u -16.0))) 8.0)) v)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - ((u * (8.0f + (u * -4.0f))) - 4.0f))) - (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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((u * (8.0e0 + (u * (-4.0e0)))) - 4.0e0))) - (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(0.11999999731779099))
		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(u * Float32(Float32(8.0) + Float32(u * Float32(-4.0)))) - Float32(4.0)))) - 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(0.11999999731779099))
		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))) - ((u * (single(8.0) + (u * single(-4.0)))) - single(4.0)))) - (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 < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around -inf 63.9%

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

   6. Taylor expanded in u around 0 63.9%

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

   7. Taylor expanded in u around 0 63.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - \left(u \cdot \left(8 + u \cdot -4\right) - 4\right)\right) - 0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 - \frac{0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot 4 + -4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    1.0
    (-
     (* (- 1.0 u) -2.0)
     (/
      (+
       (* 0.16666666666666666 (/ (* u (- (* u (+ 24.0 (* u -16.0))) 8.0)) v))
       (* -0.5 (+ (* (- 1.0 u) 4.0) (* -4.0 (* (- 1.0 u) (- 1.0 u))))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) - (((0.16666666666666666f * ((u * ((u * (24.0f + (u * -16.0f))) - 8.0f)) / v)) + (-0.5f * (((1.0f - u) * 4.0f) + (-4.0f * ((1.0f - u) * (1.0f - 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) - (((0.16666666666666666e0 * ((u * ((u * (24.0e0 + (u * (-16.0e0)))) - 8.0e0)) / v)) + ((-0.5e0) * (((1.0e0 - u) * 4.0e0) + ((-4.0e0) * ((1.0e0 - u) * (1.0e0 - u)))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		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(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))) - Float32(8.0))) / v)) + Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(1.0) - u) * Float32(4.0)) + Float32(Float32(-4.0) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) - (((single(0.16666666666666666) * ((u * ((u * (single(24.0) + (u * single(-16.0)))) - single(8.0))) / v)) + (single(-0.5) * (((single(1.0) - u) * single(4.0)) + (single(-4.0) * ((single(1.0) - u) * (single(1.0) - u)))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around -inf 63.9%

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

   6. Taylor expanded in u around 0 63.9%

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

      1. unpow2 63.9%

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

   8. Applied egg-rr 63.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 - \frac{0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot 4 + -4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(2 - u \cdot 2\right) - 0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* u (- 2.0 (* 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 <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((u * (2.0f - (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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (((u * (2.0e0 - (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(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(u * Float32(Float32(2.0) - 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(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((u * (single(2.0) - (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 < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around -inf 63.9%

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

   6. Taylor expanded in u around 0 63.9%

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

   7. Taylor expanded in u around 0 63.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(2 - u \cdot 2\right) - 0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot \left(24 + u \cdot -16\right) - 8\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    -1.0
    (- (* u 2.0) (/ (+ (* u -2.0) (* -1.3333333333333333 (/ u v))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((u * -2.0f) + (-1.3333333333333333f * (u / 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - (((u * (-2.0e0)) + ((-1.3333333333333333e0) * (u / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(-1.3333333333333333) * Float32(u / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (((u * single(-2.0)) + (single(-1.3333333333333333) * (u / v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in u around 0 57.3%

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

      1. *-commutative 57.3%

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

      2. associate-*l* 57.3%

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

      3. *-commutative 57.3%

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

      4. fmm-def 57.3%

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

      5. rec-exp 57.4%

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

      6. expm1-define 57.4%

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

      7. distribute-neg-frac 57.4%

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

      8. metadata-eval 57.4%

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

      9. metadata-eval 57.4%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in v around -inf 58.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 + -1.3333333333333333 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} + 2 \cdot \frac{1}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+ -1.0 (* u (+ 2.0 (+ (* -2.0 (/ u v)) (* 2.0 (/ 1.0 v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((-2.0f * (u / v)) + (2.0f * (1.0f / 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((-2.0e0) * (u / v)) + (2.0e0 * (1.0e0 / v)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(-2.0) * Float32(u / v)) + Float32(Float32(2.0) * Float32(Float32(1.0) / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(-2.0) * (u / v)) + (single(2.0) * (single(1.0) / v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around inf 56.9%

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

   6. Taylor expanded in u around 0 56.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} + 2 \cdot \frac{1}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 2 \cdot \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+ 1.0 (+ (* (- 1.0 u) -2.0) (* 2.0 (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (2.0f * (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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (2.0e0 * (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(2.0) * Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (single(2.0) * (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around inf 56.9%

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

   6. Taylor expanded in u around 0 55.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 2 \cdot \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - 2 \cdot \frac{-1}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099) 1.0 (+ -1.0 (* u (- 2.0 (* 2.0 (/ -1.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - (2.0f * (-1.0f / 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (2.0e0 * ((-1.0e0) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(2.0) * Float32(Float32(-1.0) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - (single(2.0) * (single(-1.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around inf 56.9%

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

   6. Taylor expanded in u around 0 55.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - 2 \cdot \frac{-1}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		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 <= 0.11999999731779099e0) 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(0.11999999731779099))
		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(0.11999999731779099))
		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 < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in u around 0 57.3%

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

      1. *-commutative 57.3%

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

      2. associate-*l* 57.3%

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

      3. *-commutative 57.3%

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

      4. fmm-def 57.3%

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

      5. rec-exp 57.4%

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

      6. expm1-define 57.4%

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

      7. distribute-neg-frac 57.4%

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

      8. metadata-eval 57.4%

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

      9. metadata-eval 57.4%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in v around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. distribute-lft-out 55.5%

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

      3. metadata-eval 55.5%

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

   10. Simplified 55.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.1% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in u around 0 57.3%

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

      1. *-commutative 57.3%

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

      2. associate-*l* 57.3%

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

      3. *-commutative 57.3%

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

      4. fmm-def 57.3%

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

      5. rec-exp 57.4%

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

      6. expm1-define 57.4%

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

      7. distribute-neg-frac 57.4%

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

      8. metadata-eval 57.4%

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

      9. metadata-eval 57.4%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in v around inf 46.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.5% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.11999999731779099) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		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 (v <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		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 (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right)} \]

      2. log1p-define 100.0%

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

      3. expm1-undefine 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

      8. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 94.1%

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

   if 0.119999997 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 92.9%

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

      3. +-commutative 92.9%

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

      4. fma-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in u around 0 38.3%

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

Alternative 15: 5.7% 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 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-define 99.3%

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

   3. +-commutative 99.3%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0 6.6%

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

Reproduce

herbie shell --seed 2024152 
(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))))))))