HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.1s

Alternatives: 22

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * (u * ((1.0e0 / u) + (-1.0e0)))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around inf 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

Alternative 5: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(v, -\log \left(\frac{1}{u}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (fma v (- (log (/ 1.0 u))) 1.0)
   (+
    -1.0
    (*
     u
     (+
      (/
       (-
        (/
         (-
          (* 0.5 (- (* u 8.0) (* u 16.0)))
          (* u (/ (+ 4.666666666666667 (/ 4.0 v)) v)))
         v)
        (* u 2.0))
       v)
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = fmaf(v, -logf((1.0f / u)), 1.0f);
	} else {
		tmp = -1.0f + (u * ((((((0.5f * ((u * 8.0f) - (u * 16.0f))) - (u * ((4.666666666666667f + (4.0f / v)) / v))) / v) - (u * 2.0f)) / v) + (v * (-1.0f + (1.0f / expf((-2.0f / v)))))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = fma(v, Float32(-log(Float32(Float32(1.0) / u))), Float32(1.0));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) - Float32(u * Float32(Float32(Float32(4.666666666666667) + Float32(Float32(4.0) / v)) / v))) / v) - Float32(u * Float32(2.0))) / v) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      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 99.7%

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

   8. Taylor expanded in u around inf 99.5%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 90.0%

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

   7. Taylor expanded in u around 0 90.0%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{-1 \cdot \frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   8. Step-by-step derivation

      1. mul-1-neg 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\left(-\frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      2. associate-/l* 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-\color{blue}{u \cdot \frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      3. distribute-rgt-neg-in 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(-\frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      4. associate-*r/ 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(-\frac{4.666666666666667 + \color{blue}{\frac{4 \cdot 1}{v}}}{v}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      5. metadata-eval 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(-\frac{4.666666666666667 + \frac{\color{blue}{4}}{v}}{v}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   9. Simplified 90.0%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(-\frac{4.666666666666667 + \frac{4}{v}}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(v, -\log \left(\frac{1}{u}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% 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 99.5%

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

3. Taylor expanded in u around 0 96.6%

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

Alternative 7: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (fma v (log u) 1.0)
   (+
    -1.0
    (*
     u
     (+
      (/
       (-
        (/
         (-
          (* 0.5 (- (* u 8.0) (* u 16.0)))
          (* u (/ (+ 4.666666666666667 (/ 4.0 v)) v)))
         v)
        (* u 2.0))
       v)
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = fmaf(v, logf(u), 1.0f);
	} else {
		tmp = -1.0f + (u * ((((((0.5f * ((u * 8.0f) - (u * 16.0f))) - (u * ((4.666666666666667f + (4.0f / v)) / v))) / v) - (u * 2.0f)) / v) + (v * (-1.0f + (1.0f / expf((-2.0f / v)))))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = fma(v, log(u), Float32(1.0));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) - Float32(u * Float32(Float32(Float32(4.666666666666667) + Float32(Float32(4.0) / v)) / v))) / v) - Float32(u * Float32(2.0))) / v) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      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 99.7%

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

   8. Taylor expanded in u around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. log-rec 99.5%

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

      3. remove-double-neg 99.5%

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

   10. Simplified 99.5%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 90.0%

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

   7. Taylor expanded in u around 0 90.0%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{-1 \cdot \frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   8. Step-by-step derivation

      1. mul-1-neg 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\left(-\frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      2. associate-/l* 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-\color{blue}{u \cdot \frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      3. distribute-rgt-neg-in 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(-\frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      4. associate-*r/ 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(-\frac{4.666666666666667 + \color{blue}{\frac{4 \cdot 1}{v}}}{v}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      5. metadata-eval 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(-\frac{4.666666666666667 + \frac{\color{blue}{4}}{v}}{v}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   9. Simplified 90.0%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(-\frac{4.666666666666667 + \frac{4}{v}}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+
    -1.0
    (*
     u
     (+
      (/
       (-
        (/
         (-
          (* 0.5 (- (* u 8.0) (* u 16.0)))
          (* u (/ (+ 4.666666666666667 (/ 4.0 v)) v)))
         v)
        (* u 2.0))
       v)
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = -1.0f + (u * ((((((0.5f * ((u * 8.0f) - (u * 16.0f))) - (u * ((4.666666666666667f + (4.0f / v)) / v))) / v) - (u * 2.0f)) / v) + (v * (-1.0f + (1.0f / expf((-2.0f / v)))))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.5e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = (-1.0e0) + (u * ((((((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) - (u * ((4.666666666666667e0 + (4.0e0 / v)) / v))) / v) - (u * 2.0e0)) / v) + (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v)))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) - Float32(u * Float32(Float32(Float32(4.666666666666667) + Float32(Float32(4.0) / v)) / v))) / v) - Float32(u * Float32(2.0))) / v) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(-1.0) + (u * ((((((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) - (u * ((single(4.666666666666667) + (single(4.0) / v)) / v))) / v) - (u * single(2.0))) / v) + (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      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 99.7%

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

   8. Taylor expanded in u around inf 99.5%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 90.0%

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

   7. Taylor expanded in u around 0 90.0%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{-1 \cdot \frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   8. Step-by-step derivation

      1. mul-1-neg 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\left(-\frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      2. associate-/l* 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-\color{blue}{u \cdot \frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      3. distribute-rgt-neg-in 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(-\frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      4. associate-*r/ 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(-\frac{4.666666666666667 + \color{blue}{\frac{4 \cdot 1}{v}}}{v}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      5. metadata-eval 90.0%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(-\frac{4.666666666666667 + \frac{\color{blue}{4}}{v}}{v}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   9. Simplified 90.0%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(-\frac{4.666666666666667 + \frac{4}{v}}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / 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 <= 0.5e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / 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(0.5))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / 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(-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(0.5))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / 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 < 0.5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      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 99.7%

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

   8. Taylor expanded in u around inf 99.5%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. distribute-neg-frac2 87.4%

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

      3. associate--l+ 87.4%

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

      4. *-commutative 87.4%

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

      5. associate-*r/ 87.4%

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

      6. associate-*r/ 87.4%

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

      7. div-sub 87.4%

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

      8. distribute-rgt-out-- 87.4%

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

      9. metadata-eval 87.4%

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

      10. *-commutative 87.4%

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

      11. associate-*r/ 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 98.9%

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

Alternative 10: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot \left(0.6666666666666666 - u \cdot 4.666666666666667\right)}{v} + u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (/
        (+
         (/ (* u (- 0.6666666666666666 (* u 4.666666666666667))) v)
         (* u (- 1.3333333333333333 (* 0.5 (- (* u 16.0) (* u 8.0))))))
        v)
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = -1.0f + ((u * 2.0f) + ((((((u * (0.6666666666666666f - (u * 4.666666666666667f))) / v) + (u * (1.3333333333333333f - (0.5f * ((u * 16.0f) - (u * 8.0f)))))) / v) + (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 <= 0.5e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((((((u * (0.6666666666666666e0 - (u * 4.666666666666667e0))) / v) + (u * (1.3333333333333333e0 - (0.5e0 * ((u * 16.0e0) - (u * 8.0e0)))))) / v) + (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(0.5))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(Float32(0.6666666666666666) - Float32(u * Float32(4.666666666666667)))) / v) + Float32(u * Float32(Float32(1.3333333333333333) - Float32(Float32(0.5) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))))))) / v) + 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(0.5))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(-1.0) + ((u * single(2.0)) + ((((((u * (single(0.6666666666666666) - (u * single(4.666666666666667)))) / v) + (u * (single(1.3333333333333333) - (single(0.5) * ((u * single(16.0)) - (u * single(8.0))))))) / v) + (u * (single(2.0) - (u * single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      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 99.7%

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

   8. Taylor expanded in u around inf 99.5%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 87.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 87.4%

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(4.666666666666667 \cdot u - 0.6666666666666666\right)}}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot \left(0.6666666666666666 - u \cdot 4.666666666666667\right)}{v} + u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot \left(0.6666666666666666 - u \cdot 4.666666666666667\right)}{v} + u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (/
        (+
         (/ (* u (- 0.6666666666666666 (* u 4.666666666666667))) v)
         (* u (- 1.3333333333333333 (* 0.5 (- (* u 16.0) (* u 8.0))))))
        v)
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + ((u * 2.0f) + ((((((u * (0.6666666666666666f - (u * 4.666666666666667f))) / v) + (u * (1.3333333333333333f - (0.5f * ((u * 16.0f) - (u * 8.0f)))))) / v) + (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 <= 0.5e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((((((u * (0.6666666666666666e0 - (u * 4.666666666666667e0))) / v) + (u * (1.3333333333333333e0 - (0.5e0 * ((u * 16.0e0) - (u * 8.0e0)))))) / v) + (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(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(Float32(0.6666666666666666) - Float32(u * Float32(4.666666666666667)))) / v) + Float32(u * Float32(Float32(1.3333333333333333) - Float32(Float32(0.5) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))))))) / v) + 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(0.5))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + ((u * single(2.0)) + ((((((u * (single(0.6666666666666666) - (u * single(4.666666666666667)))) / v) + (u * (single(1.3333333333333333) - (single(0.5) * ((u * single(16.0)) - (u * single(8.0))))))) / v) + (u * (single(2.0) - (u * single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      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 99.7%

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

   8. Taylor expanded in u around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. log-rec 99.5%

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

      4. remove-double-neg 99.5%

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

   10. Simplified 99.5%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 87.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 87.4%

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(4.666666666666667 \cdot u - 0.6666666666666666\right)}}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot \left(0.6666666666666666 - u \cdot 4.666666666666667\right)}{v} + u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot \left(0.6666666666666666 - u \cdot 4.666666666666667\right)}{v} + u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (/
        (+
         (/ (* u (- 0.6666666666666666 (* u 4.666666666666667))) v)
         (* u (- 1.3333333333333333 (* 0.5 (- (* u 16.0) (* u 8.0))))))
        v)
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + ((((((u * (0.6666666666666666f - (u * 4.666666666666667f))) / v) + (u * (1.3333333333333333f - (0.5f * ((u * 16.0f) - (u * 8.0f)))))) / v) + (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((((((u * (0.6666666666666666e0 - (u * 4.666666666666667e0))) / v) + (u * (1.3333333333333333e0 - (0.5e0 * ((u * 16.0e0) - (u * 8.0e0)))))) / v) + (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(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(Float32(0.6666666666666666) - Float32(u * Float32(4.666666666666667)))) / v) + Float32(u * Float32(Float32(1.3333333333333333) - Float32(Float32(0.5) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))))))) / v) + 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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + ((((((u * (single(0.6666666666666666) - (u * single(4.666666666666667)))) / v) + (u * (single(1.3333333333333333) - (single(0.5) * ((u * single(16.0)) - (u * single(8.0))))))) / v) + (u * (single(2.0) - (u * single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.200000003 < v

   1. Initial program 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-define 91.7%

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

      3. +-commutative 91.7%

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

      4. fma-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in u around 0 83.2%

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

   6. Taylor expanded in v around -inf 82.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 82.9%

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\color{blue}{u \cdot \left(4.666666666666667 \cdot u - 0.6666666666666666\right)}}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot \left(0.6666666666666666 - u \cdot 4.666666666666667\right)}{v} + u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) + \frac{u}{v} \cdot 0.6666666666666666}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (/
        (+
         (* u (- 1.3333333333333333 (* 0.5 (- (* u 16.0) (* u 8.0)))))
         (* (/ u v) 0.6666666666666666))
        v)
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + (((((u * (1.3333333333333333f - (0.5f * ((u * 16.0f) - (u * 8.0f))))) + ((u / v) * 0.6666666666666666f)) / v) + (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 <= 0.5e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + (((((u * (1.3333333333333333e0 - (0.5e0 * ((u * 16.0e0) - (u * 8.0e0))))) + ((u / v) * 0.6666666666666666e0)) / v) + (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(0.5))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(Float32(u * Float32(Float32(1.3333333333333333) - Float32(Float32(0.5) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))))) + Float32(Float32(u / v) * Float32(0.6666666666666666))) / v) + 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(0.5))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + (((((u * (single(1.3333333333333333) - (single(0.5) * ((u * single(16.0)) - (u * single(8.0)))))) + ((u / v) * single(0.6666666666666666))) / v) + (u * (single(2.0) - (u * single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 87.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 86.2%

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{0.6666666666666666 \cdot \frac{u}{v}} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1 \]
   8. Step-by-step derivation

      1. *-commutative 86.2%

         \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\frac{u}{v} \cdot 0.6666666666666666} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1 \]

   9. Simplified 86.2%

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\frac{u}{v} \cdot 0.6666666666666666} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{u \cdot \left(1.3333333333333333 - 0.5 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) + \frac{u}{v} \cdot 0.6666666666666666}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-2 \cdot \frac{u}{v} + v \cdot \left(-1 + \left(1 - \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (* -2.0 (/ u v))
      (*
       v
       (+
        -1.0
        (- 1.0 (/ (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v)) v)))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((-2.0f * (u / v)) + (v * (-1.0f + (1.0f - ((-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (((-2.0e0) * (u / v)) + (v * ((-1.0e0) + (1.0e0 - (((-2.0e0) + (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)) / v))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-2.0) * Float32(u / v)) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(-2.0) * (u / v)) + (v * (single(-1.0) + (single(1.0) - ((single(-2.0) + ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)) / v))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.4%

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

   if 0.150000006 < v

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.3%

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

      3. +-commutative 92.3%

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

      4. fma-define 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in u around 0 77.9%

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

   6. Taylor expanded in v around inf 74.2%

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

   7. Taylor expanded in v around -inf 74.0%

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

      1. mul-1-neg 74.0%

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

      2. unsub-neg 74.0%

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

      3. sub-neg 74.0%

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

      4. associate-*r/ 74.0%

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

      5. neg-mul-1 74.0%

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

      6. distribute-neg-in 74.0%

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

      7. metadata-eval 74.0%

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

      8. associate-*r/ 74.0%

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

      9. metadata-eval 74.0%

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

      10. distribute-neg-frac 74.0%

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

      11. metadata-eval 74.0%

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

      12. metadata-eval 74.0%

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

   9. Simplified 74.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-2 \cdot \frac{u}{v} + v \cdot \left(-1 + \left(1 - \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.3% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(-2.0) * (u / v)) + (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around inf 82.8%

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

   7. Taylor expanded in v around -inf 82.1%

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

4. Final simplification 91.4%

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

Alternative 16: 91.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-2 \cdot \frac{u}{v} - v \cdot \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (*
     u
     (-
      (* -2.0 (/ u v))
      (* v (/ (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((-2.0f * (u / v)) - (v * ((-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (((-2.0e0) * (u / v)) - (v * (((-2.0e0) + (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-2.0) * Float32(u / v)) - Float32(v * Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(-2.0) * (u / v)) - (v * ((single(-2.0) + ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.4%

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

   if 0.150000006 < v

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.3%

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

      3. +-commutative 92.3%

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

      4. fma-define 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in u around 0 77.9%

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

   6. Taylor expanded in v around inf 74.2%

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

   7. Taylor expanded in v around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. distribute-neg-frac2 73.9%

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

      3. sub-neg 73.9%

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

      4. associate-*r/ 73.9%

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

      5. distribute-lft-in 73.9%

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

      6. metadata-eval 73.9%

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

      7. neg-mul-1 73.9%

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

      8. associate-*r/ 73.9%

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

      9. metadata-eval 73.9%

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

      10. distribute-neg-frac 73.9%

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

      11. metadata-eval 73.9%

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

      12. metadata-eval 73.9%

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

   9. Simplified 73.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-2 \cdot \frac{u}{v} - v \cdot \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 91.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(2 - u \cdot 2\right) - \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/ (- (* u (- 2.0 (* u 2.0))) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + (((u * (2.0f - (u * 2.0f))) - ((u / v) * -1.3333333333333333f)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + (((u * (2.0e0 - (u * 2.0e0))) - ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(u * Float32(Float32(2.0) - Float32(u * Float32(2.0)))) - Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + (((u * (single(2.0) - (u * single(2.0)))) - ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.4%

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

   if 0.150000006 < v

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.3%

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

      3. +-commutative 92.3%

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

      4. fma-define 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in u around 0 77.9%

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

   6. Taylor expanded in v around -inf 74.4%

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

   7. Taylor expanded in u around 0 73.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(2 - u \cdot 2\right) - \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 91.2% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(-2 + \frac{-1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   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 <= 0.15000000596046448f) {
		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 <= 0.15000000596046448e0) 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(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(u * 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(0.15000000596046448))
		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 < 0.150000006

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.4%

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

   if 0.150000006 < v

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.3%

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

      3. +-commutative 92.3%

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

      4. fma-define 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in u around 0 77.9%

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

   6. Taylor expanded in v around -inf 74.4%

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

   7. Taylor expanded in u around 0 71.0%

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

      1. mul-1-neg 71.0%

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

      2. distribute-rgt-neg-in 71.0%

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

      3. distribute-neg-in 71.0%

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

      4. metadata-eval 71.0%

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

      5. associate-*r/ 71.0%

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

      6. metadata-eval 71.0%

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

      7. distribute-neg-frac 71.0%

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

      8. metadata-eval 71.0%

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

   9. Simplified 71.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(-2 + \frac{-1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 90.8% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around inf 82.8%

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

   7. Taylor expanded in v around inf 70.7%

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

      1. associate-*r/ 70.7%

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

      2. metadata-eval 70.7%

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

      3. metadata-eval 70.7%

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

      4. distribute-neg-frac 70.7%

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

      5. unsub-neg 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 90.8%

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

Alternative 20: 90.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		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 <= 0.5e0) 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(0.5))
		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(0.5))
		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 < 0.5

   1. Initial program 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.5 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. fma-define 91.5%

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

      3. +-commutative 91.5%

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

      4. fma-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around inf 82.8%

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

   7. Taylor expanded in v around inf 70.7%

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

      1. +-commutative 70.7%

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

      2. associate-/l* 70.7%

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

      3. *-commutative 70.7%

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

      4. distribute-lft-out 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 90.8%

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

Alternative 21: 87.0% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. log1p-expm1-u 99.5%

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

   2. expm1-undefine 99.4%

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

   3. *-commutative 99.4%

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

   4. exp-to-pow 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in v around 0 87.4%

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

Alternative 22: 6.0% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 5.5%

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

Reproduce

herbie shell --seed 2024169 
(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))))))))