HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 16.7s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(Float32(log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u)) * sqrt(v)), sqrt(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. Add Preprocessing
3. 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. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.4%

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

   4. associate-*r* 99.5%

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

   5. fma-define 99.5%

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

   6. +-commutative 99.5%

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

   7. fma-undefine 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 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. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
   (- -1.0 (* u (* v (- (/ -1.0 (exp (/ -2.0 v))) -1.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = -1.0f - (u * (v * ((-1.0f / expf((-2.0f / v))) - -1.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in u around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. distribute-lft-neg-in 99.8%

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

      9. *-commutative 99.8%

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

      10. expm1-define 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in v around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. expm1-define 99.7%

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

      3. *-commutative 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   10. Simplified 99.7%

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

   if 0.219999999 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in u around 0 71.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. Final simplification 99.5%

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

Alternative 6: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   1.0
   (- -1.0 (* u (* v (- (/ -1.0 (exp (/ -2.0 v))) -1.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f - (u * (v * ((-1.0f / expf((-2.0f / v))) - -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) - (u * (v * (((-1.0e0) / exp(((-2.0e0) / v))) - (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.2199999988079071))
		tmp = single(1.0);
	else
		tmp = single(-1.0) - (u * (v * ((single(-1.0) / exp((single(-2.0) / v))) - single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.1%

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

   if 0.219999999 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in u around 0 71.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   1.0
   (+ 1.0 (- (* u (* v (expm1 (/ 2.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * expm1f((2.0f / v)))) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.1%

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

   if 0.219999999 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in u around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. sub-neg 71.2%

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

      3. metadata-eval 71.2%

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

      4. distribute-lft-in 71.2%

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

      5. rec-exp 71.2%

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

      6. distribute-neg-frac 71.2%

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

      7. metadata-eval 71.2%

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

   5. Applied egg-rr 71.2%

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

      1. distribute-lft-out 71.2%

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

      2. metadata-eval 71.2%

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

      3. associate-*l/ 71.2%

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

      4. exp-prod 71.2%

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

      5. unpow-1 71.2%

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

      6. metadata-eval 71.2%

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

      7. sub-neg 71.2%

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

      8. associate-*r* 71.2%

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

      9. rec-exp 71.2%

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

      10. expm1-define 71.2%

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

      11. distribute-neg-frac 71.2%

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

      12. metadata-eval 71.2%

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

   7. Simplified 71.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((2.0f - ((-2.0f + ((-1.3333333333333333f + (-0.6666666666666666f / v)) / v)) / v)) + (-1.0f / u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = u * ((2.0e0 - (((-2.0e0) + (((-1.3333333333333333e0) + ((-0.6666666666666666e0) / v)) / v)) / v)) + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(2.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-1.3333333333333333) + Float32(Float32(-0.6666666666666666) / v)) / v)) / v)) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = u * ((single(2.0) - ((single(-2.0) + ((single(-1.3333333333333333) + (single(-0.6666666666666666) / v)) / v)) / v)) + (single(-1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in u around 0 67.6%

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

   4. Taylor expanded in v around -inf 64.1%

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

   5. Taylor expanded in u around 0 64.1%

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

      1. associate-/l* 64.1%

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

      2. associate-*r/ 64.1%

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

      3. metadata-eval 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in u around inf 64.1%

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

      1. sub-neg 64.1%

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

   10. Simplified 64.1%

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

4. Final simplification 91.3%

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

Alternative 9: 90.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = ((u * 2.0f) + ((u * (((1.3333333333333333f + (0.6666666666666666f / v)) / v) - -2.0f)) / v)) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = ((u * 2.0e0) + ((u * (((1.3333333333333333e0 + (0.6666666666666666e0 / v)) / v) - (-2.0e0))) / v)) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(u * Float32(2.0)) + Float32(Float32(u * Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v) - Float32(-2.0))) / v)) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = ((u * single(2.0)) + ((u * (((single(1.3333333333333333) + (single(0.6666666666666666) / v)) / v) - single(-2.0))) / v)) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in u around 0 67.6%

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

   4. Taylor expanded in v around -inf 64.1%

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

   5. Taylor expanded in u around 0 64.1%

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

      1. associate-/l* 64.1%

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

      2. associate-*r/ 64.1%

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

      3. metadata-eval 64.1%

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

   7. Simplified 64.1%

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

      1. associate-+r- 64.1%

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

   9. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(u, 2, -\frac{u \cdot -2 - \frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}}{v}\right)\right) - 2} \]
   10. Step-by-step derivation

       1. sub-neg 64.1%

          \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(u, 2, -\frac{u \cdot -2 - \frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}}{v}\right)\right) + \left(-2\right)} \]

       2. +-commutative 64.1%

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u, 2, -\frac{u \cdot -2 - \frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}}{v}\right) + 1\right)} + \left(-2\right) \]

       3. metadata-eval 64.1%

          \[\leadsto \left(\mathsf{fma}\left(u, 2, -\frac{u \cdot -2 - \frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}}{v}\right) + 1\right) + \color{blue}{-2} \]

       4. associate-+l+ 64.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(u, 2, -\frac{u \cdot -2 - \frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}}{v}\right) + \left(1 + -2\right)} \]

       5. fmm-undef 64.3%

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

       6. associate-/l* 64.3%

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

       7. distribute-lft-out-- 64.3%

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

       8. metadata-eval 64.3%

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

   11. Simplified 64.3%

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

4. Final simplification 91.3%

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

Alternative 10: 90.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf 60.7%

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

      1. fma-define 60.7%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 60.7%

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

      5. *-commutative 60.7%

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

      6. unpow2 60.7%

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

      7. associate-*l* 60.7%

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

      8. *-commutative 60.7%

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

      9. distribute-lft-out 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in u around 0 60.7%

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

4. Final simplification 91.0%

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

Alternative 11: 90.4% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf 60.7%

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

      1. fma-define 60.7%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 60.7%

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

      5. *-commutative 60.7%

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

      6. unpow2 60.7%

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

      7. associate-*l* 60.7%

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

      8. *-commutative 60.7%

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

      9. distribute-lft-out 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in u around 0 57.7%

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

4. Final simplification 90.8%

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

Alternative 12: 90.4% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in u around 0 67.6%

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

   4. Taylor expanded in v around -inf 64.1%

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

   5. Taylor expanded in u around 0 64.1%

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

      1. associate-/l* 64.1%

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

      2. associate-*r/ 64.1%

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

      3. metadata-eval 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in v around inf 57.7%

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

      1. sub-neg 57.7%

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

      2. distribute-lft-out 57.7%

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

      3. metadata-eval 57.7%

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

   10. Simplified 57.7%

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

4. Final simplification 90.8%

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

Alternative 13: 89.8% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf 51.5%

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

4. Final simplification 90.3%

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

Alternative 14: 89.8% accurate, 21.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.4%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-define 93.2%

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

      3. +-commutative 93.2%

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

      4. fma-define 93.4%

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

   3. Simplified 93.4%

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

      1. fma-undefine 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in v around inf 51.5%

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

   8. Taylor expanded in u around 0 51.5%

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

4. Final simplification 90.3%

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

Alternative 15: 5.8% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 6.1%

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

4. Final simplification 6.1%

   \[\leadsto -1 \]
5. Add Preprocessing

Alternative 16: 86.8% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.4%

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

   4. associate-*r* 99.5%

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

   5. fma-define 99.5%

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

   6. +-commutative 99.5%

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

   7. fma-undefine 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in v around 0 86.9%

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

6. Final simplification 86.9%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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))))))))