HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.7s

Alternatives: 16

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, 0.7× 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 2: 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 3: 95.8% 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 99.6%

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

3. Taylor expanded in u around 0 96.5%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 8 - u \cdot 16\\ t_1 := 4 \cdot t\_0 + u \cdot 32\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 - \frac{\frac{-0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 - t\_1\right) + \left(8 \cdot t\_0 + u \cdot 42.666666666666664\right)\right) - u \cdot 8}{v} + 0.5 \cdot \left(t\_1 - u \cdot 9.333333333333334\right)}{v} + t\_0 \cdot 0.5}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 8.0) (* u 16.0))) (t_1 (+ (* 4.0 t_0) (* u 32.0))))
   (if (<= v 0.10000000149011612)
     (fma v (log u) 1.0)
     (+
      -1.0
      (*
       u
       (-
        (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
        (/
         (-
          (* u 2.0)
          (/
           (+
            (/
             (+
              (*
               -0.5
               (/
                (-
                 (+
                  (* 4.0 (- (* u 9.333333333333334) t_1))
                  (+ (* 8.0 t_0) (* u 42.666666666666664)))
                 (* u 8.0))
                v))
              (* 0.5 (- t_1 (* u 9.333333333333334))))
             v)
            (* t_0 0.5))
           v))
         v)))))))

C

float code(float u, float v) {
	float t_0 = (u * 8.0f) - (u * 16.0f);
	float t_1 = (4.0f * t_0) + (u * 32.0f);
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = fmaf(v, logf(u), 1.0f);
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (((u * 2.0f) - (((((-0.5f * ((((4.0f * ((u * 9.333333333333334f) - t_1)) + ((8.0f * t_0) + (u * 42.666666666666664f))) - (u * 8.0f)) / v)) + (0.5f * (t_1 - (u * 9.333333333333334f)))) / v) + (t_0 * 0.5f)) / v)) / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))
	t_1 = Float32(Float32(Float32(4.0) * t_0) + Float32(u * Float32(32.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = fma(v, log(u), Float32(1.0));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) - Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(9.333333333333334)) - t_1)) + Float32(Float32(Float32(8.0) * t_0) + Float32(u * Float32(42.666666666666664)))) - Float32(u * Float32(8.0))) / v)) + Float32(Float32(0.5) * Float32(t_1 - Float32(u * Float32(9.333333333333334))))) / v) + Float32(t_0 * Float32(0.5))) / v)) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 100.0%

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

   8. Taylor expanded in u around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. remove-double-neg 100.0%

         \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around -inf 81.0%

      \[\leadsto u \cdot \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.5 \cdot \frac{8 \cdot u - \left(4 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + \left(8 \cdot \left(8 \cdot u - 16 \cdot u\right) + 42.666666666666664 \cdot u\right)\right)}{v} + 0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right)}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 - \frac{\frac{-0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 - \left(4 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 32\right)\right) + \left(8 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 42.666666666666664\right)\right) - u \cdot 8}{v} + 0.5 \cdot \left(\left(4 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 32\right) - u \cdot 9.333333333333334\right)}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 8 - u \cdot 16\\ t_1 := 4 \cdot t\_0 + u \cdot 32\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 - \frac{\frac{-0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 - t\_1\right) + \left(8 \cdot t\_0 + u \cdot 42.666666666666664\right)\right) - u \cdot 8}{v} + 0.5 \cdot \left(t\_1 - u \cdot 9.333333333333334\right)}{v} + t\_0 \cdot 0.5}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 8.0) (* u 16.0))) (t_1 (+ (* 4.0 t_0) (* u 32.0))))
   (if (<= v 0.10000000149011612)
     (+ 1.0 (* v (log u)))
     (+
      -1.0
      (*
       u
       (-
        (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
        (/
         (-
          (* u 2.0)
          (/
           (+
            (/
             (+
              (*
               -0.5
               (/
                (-
                 (+
                  (* 4.0 (- (* u 9.333333333333334) t_1))
                  (+ (* 8.0 t_0) (* u 42.666666666666664)))
                 (* u 8.0))
                v))
              (* 0.5 (- t_1 (* u 9.333333333333334))))
             v)
            (* t_0 0.5))
           v))
         v)))))))

C

float code(float u, float v) {
	float t_0 = (u * 8.0f) - (u * 16.0f);
	float t_1 = (4.0f * t_0) + (u * 32.0f);
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (((u * 2.0f) - (((((-0.5f * ((((4.0f * ((u * 9.333333333333334f) - t_1)) + ((8.0f * t_0) + (u * 42.666666666666664f))) - (u * 8.0f)) / v)) + (0.5f * (t_1 - (u * 9.333333333333334f)))) / v) + (t_0 * 0.5f)) / v)) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (u * 8.0e0) - (u * 16.0e0)
    t_1 = (4.0e0 * t_0) + (u * 32.0e0)
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) - (((u * 2.0e0) - ((((((-0.5e0) * ((((4.0e0 * ((u * 9.333333333333334e0) - t_1)) + ((8.0e0 * t_0) + (u * 42.666666666666664e0))) - (u * 8.0e0)) / v)) + (0.5e0 * (t_1 - (u * 9.333333333333334e0)))) / v) + (t_0 * 0.5e0)) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))
	t_1 = Float32(Float32(Float32(4.0) * t_0) + Float32(u * Float32(32.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) - Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(9.333333333333334)) - t_1)) + Float32(Float32(Float32(8.0) * t_0) + Float32(u * Float32(42.666666666666664)))) - Float32(u * Float32(8.0))) / v)) + Float32(Float32(0.5) * Float32(t_1 - Float32(u * Float32(9.333333333333334))))) / v) + Float32(t_0 * Float32(0.5))) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (u * single(8.0)) - (u * single(16.0));
	t_1 = (single(4.0) * t_0) + (u * single(32.0));
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) - (((u * single(2.0)) - (((((single(-0.5) * ((((single(4.0) * ((u * single(9.333333333333334)) - t_1)) + ((single(8.0) * t_0) + (u * single(42.666666666666664)))) - (u * single(8.0))) / v)) + (single(0.5) * (t_1 - (u * single(9.333333333333334))))) / v) + (t_0 * single(0.5))) / 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 100.0%

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

   8. Taylor expanded in u around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around -inf 81.0%

      \[\leadsto u \cdot \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.5 \cdot \frac{8 \cdot u - \left(4 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + \left(8 \cdot \left(8 \cdot u - 16 \cdot u\right) + 42.666666666666664 \cdot u\right)\right)}{v} + 0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right)}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 - \frac{\frac{-0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 - \left(4 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 32\right)\right) + \left(8 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 42.666666666666664\right)\right) - u \cdot 8}{v} + 0.5 \cdot \left(\left(4 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 32\right) - u \cdot 9.333333333333334\right)}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (-
      (* v (expm1 (/ 2.0 v)))
      (* -0.5 (/ (+ (* u -4.0) (* -8.0 (/ u v))) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((v * expm1f((2.0f / v))) - (-0.5f * (((u * -4.0f) + (-8.0f * (u / v))) / v))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(v * expm1(Float32(Float32(2.0) / v))) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(-8.0) * Float32(u / v))) / v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 100.0%

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

   8. Taylor expanded in u around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around -inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

      3. associate--l+ 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*r/ 80.2%

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

      6. associate-*r/ 80.2%

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

      7. div-sub 80.2%

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

      8. distribute-rgt-out-- 80.2%

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

      9. metadata-eval 80.2%

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

      10. *-commutative 80.2%

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

      11. associate-*r/ 80.2%

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

   8. Simplified 80.2%

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

      1. pow1 80.2%

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

      2. rec-exp 80.2%

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

      3. expm1-define 80.2%

         \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + {\left(v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right)}^{1}\right) - 1 \]

   10. Applied egg-rr 80.2%

       \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + \color{blue}{{\left(v \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)}^{1}}\right) - 1 \]
   11. Step-by-step derivation

       1. unpow1 80.2%

          \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + \color{blue}{v \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) - 1 \]

       2. distribute-neg-frac 80.2%

          \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) - 1 \]

       3. metadata-eval 80.2%

          \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) - 1 \]

   12. Simplified 80.2%

       \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + \color{blue}{v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (-
      (+
       2.0
       (/
        (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) v))
        v))
      (* -0.5 (/ (+ (* u -4.0) (* -8.0 (/ u v))) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f * (1.0f / v))) / v)) / v)) - (-0.5f * (((u * -4.0f) + (-8.0f * (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * ((2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v)) - ((-0.5e0) * (((u * (-4.0e0)) + ((-8.0e0) * (u / v))) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v)) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(-8.0) * Float32(u / 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) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v)) - (single(-0.5) * (((u * single(-4.0)) + (single(-8.0) * (u / 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 100.0%

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

   8. Taylor expanded in u around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around -inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

      3. associate--l+ 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*r/ 80.2%

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

      6. associate-*r/ 80.2%

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

      7. div-sub 80.2%

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

      8. distribute-rgt-out-- 80.2%

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

      9. metadata-eval 80.2%

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

      10. *-commutative 80.2%

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

      11. associate-*r/ 80.2%

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

   8. Simplified 80.2%

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

   9. Taylor expanded in v around -inf 76.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (-
      (+
       2.0
       (/
        (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) v))
        v))
      (* -0.5 (/ (+ (* u -4.0) (* -8.0 (/ u v))) 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 + (0.6666666666666666f * (1.0f / v))) / v)) / v)) - (-0.5f * (((u * -4.0f) + (-8.0f * (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v)) - ((-0.5e0) * (((u * (-4.0e0)) + ((-8.0e0) * (u / 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(u * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v)) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(-8.0) * Float32(u / 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) + (u * ((single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v)) - (single(-0.5) * (((u * single(-4.0)) + (single(-8.0) * (u / 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. 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 94.0%

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

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around -inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

      3. associate--l+ 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*r/ 80.2%

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

      6. associate-*r/ 80.2%

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

      7. div-sub 80.2%

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

      8. distribute-rgt-out-- 80.2%

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

      9. metadata-eval 80.2%

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

      10. *-commutative 80.2%

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

      11. associate-*r/ 80.2%

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

   8. Simplified 80.2%

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

   9. Taylor expanded in v around -inf 76.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v} + \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{\left(u \cdot 2 - \frac{u \cdot \left(-4 + \frac{1.3333333333333333}{u}\right)}{v}\right) - 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.07999999821186066)
   1.0
   (+
    -1.0
    (*
     u
     (-
      2.0
      (/
       (- (- (* u 2.0) (/ (* u (+ -4.0 (/ 1.3333333333333333 u))) v)) 2.0)
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.07999999821186066f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - ((((u * 2.0f) - ((u * (-4.0f + (1.3333333333333333f / u))) / v)) - 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 <= 0.07999999821186066e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - ((((u * 2.0e0) - ((u * ((-4.0e0) + (1.3333333333333333e0 / u))) / v)) - 2.0e0) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.07999999821186066))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(u * Float32(Float32(-4.0) + Float32(Float32(1.3333333333333333) / u))) / v)) - Float32(2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.07999999821186066))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((((u * single(2.0)) - ((u * (single(-4.0) + (single(1.3333333333333333) / u))) / v)) - single(2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0799999982

   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 94.3%

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

   if 0.0799999982 < v

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.5%

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

      3. +-commutative 94.5%

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

      4. fma-define 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in u around 0 78.2%

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

   6. Taylor expanded in v around -inf 71.5%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   7. Taylor expanded in u around inf 71.5%

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

      1. sub-neg 71.5%

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

      2. associate-*r/ 71.5%

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

      3. metadata-eval 71.5%

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

      4. metadata-eval 71.5%

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

   9. Simplified 71.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{\left(u \cdot 2 - \frac{u \cdot \left(-4 + \frac{1.3333333333333333}{u}\right)}{v}\right) - 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - \left(u \cdot 2 - \frac{u \cdot -4 + 1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.07999999821186066)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/ (- 2.0 (- (* u 2.0) (/ (+ (* u -4.0) 1.3333333333333333) v))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.07999999821186066f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f - ((u * 2.0f) - (((u * -4.0f) + 1.3333333333333333f) / 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 <= 0.07999999821186066e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 - ((u * 2.0e0) - (((u * (-4.0e0)) + 1.3333333333333333e0) / v))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.07999999821186066))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) - Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-4.0)) + Float32(1.3333333333333333)) / v))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.07999999821186066))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) - ((u * single(2.0)) - (((u * single(-4.0)) + single(1.3333333333333333)) / v))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0799999982

   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 94.3%

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

   if 0.0799999982 < v

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.5%

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

      3. +-commutative 94.5%

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

      4. fma-define 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in u around 0 78.2%

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

   6. Taylor expanded in v around -inf 71.5%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   7. Taylor expanded in u around 0 71.5%

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

      1. *-commutative 71.5%

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

   9. Simplified 71.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - \left(u \cdot 2 - \frac{u \cdot -4 + 1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.07999999821186066)
   1.0
   (+
    -1.0
    (* u (+ 2.0 (/ (+ 2.0 (- (/ 1.3333333333333333 v) (* u 2.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.07999999821186066f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f / 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 <= 0.07999999821186066e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.07999999821186066))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / 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(0.07999999821186066))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0799999982

   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 94.3%

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

   if 0.0799999982 < v

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.5%

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

      3. +-commutative 94.5%

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

      4. fma-define 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in u around 0 78.2%

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

   6. Taylor expanded in v around -inf 71.5%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   7. Taylor expanded in u around 0 70.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - u \cdot \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.07999999821186066)
   1.0
   (+ -1.0 (- (* u 2.0) (* u (/ (+ -2.0 (/ -1.3333333333333333 v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.07999999821186066f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (u * ((-2.0f + (-1.3333333333333333f / 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 <= 0.07999999821186066e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - (u * (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.07999999821186066))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(u * Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.07999999821186066))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (u * ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0799999982

   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 94.3%

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

   if 0.0799999982 < v

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. fma-define 94.5%

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

      3. +-commutative 94.5%

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

      4. fma-define 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in u around 0 78.2%

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

   6. Taylor expanded in v around -inf 71.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 66.4%

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

      1. mul-1-neg 66.4%

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

      2. associate-/l* 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. mul-1-neg 66.4%

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

      5. associate-*r/ 66.4%

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

      6. distribute-lft-in 66.4%

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

      7. metadata-eval 66.4%

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

      8. neg-mul-1 66.4%

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

      9. associate-*r/ 66.4%

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

      10. metadata-eval 66.4%

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

      11. distribute-neg-frac 66.4%

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

      12. metadata-eval 66.4%

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

   9. Simplified 66.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.07999999821186066:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - u \cdot \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.5% accurate, 10.6× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (+ (* u 2.0) (/ (* u (+ 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) + ((u * (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) + ((u * (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(Float32(u * Float32(2.0)) + Float32(Float32(u * 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)) + ((u * (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. 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 94.0%

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

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around inf 65.7%

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

4. Final simplification 92.4%

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

Alternative 14: 90.5% 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. 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 94.0%

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

   if 0.100000001 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 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 83.3%

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

   6. Taylor expanded in v around -inf 65.7%

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

4. Final simplification 92.4%

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

Alternative 15: 86.6% 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. Step-by-step derivation

   1. expm1-log1p-u 94.1%

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

   2. log1p-define 94.1%

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

   3. expm1-undefine 94.1%

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

   4. add-exp-log 99.6%

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

   5. +-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-undefine 99.6%

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

   8. fma-undefine 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0 88.7%

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

Alternative 16: 5.9% 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. 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. Taylor expanded in u around 0 5.6%

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

Reproduce

herbie shell --seed 2024132 
(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))))))))