HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 27.0s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in v around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. fma-def 99.5%

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

4. Simplified 99.5%

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

   1. expm1-log1p-u 15.2%

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

   2. expm1-def 15.2%

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

   3. log1p-udef 15.3%

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

   4. add-exp-log 99.6%

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

   5. fma-udef 99.5%

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

   6. *-commutative 99.5%

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

   7. fma-def 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in v around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. fma-def 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 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-def 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. Taylor expanded in v around 0 99.5%

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

5. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * (1.0e0 - u)))))
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(Float32(1.0) - u))))))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Final simplification 99.4%

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

Alternative 5: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (* (- u) (expm1 (/ -2.0 v)))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((-u * expm1f((-2.0f / v)))));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(Float32(-u) * expm1(Float32(Float32(-2.0) / v))))))
end

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in v around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. fma-def 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in u around -inf 94.8%

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

   1. associate-*r* 94.8%

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

   2. neg-mul-1 94.8%

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

   3. expm1-def 94.8%

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

7. Simplified 94.8%

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

8. Final simplification 94.8%

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

Alternative 6: 91.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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-def 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-def 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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.119999997 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in u around 0 61.9%

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

      1. sub-neg 61.9%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in v around inf 56.6%

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

      1. fma-def 56.6%

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

      2. unpow2 56.6%

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

      3. distribute-lft-out 56.6%

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

   7. Simplified 56.6%

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

4. Final simplification 89.5%

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

Alternative 7: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \left(-1 + e^{\frac{2}{v}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ -1.0 (* v (* u (+ -1.0 (exp (/ 2.0 v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (v * (u * (-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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (v * (u * ((-1.0e0) + exp((2.0e0 / v)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(v * Float32(u * 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(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (v * (u * (single(-1.0) + exp((single(2.0) / v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.25 < v

   1. Initial program 92.1%

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

   2. Taylor expanded in u around 0 71.0%

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

      1. sub-neg 71.0%

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

   4. Simplified 71.1%

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

   5. Taylor expanded in u around 0 64.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \left(-1 + e^{\frac{2}{v}}\right)\right)\\ \end{array} \]

Alternative 8: 91.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ -1.0 (* (* v u) (expm1 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((v * u) * expm1f((2.0f / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(v * u) * expm1(Float32(Float32(2.0) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.25 < v

   1. Initial program 92.1%

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

   2. Taylor expanded in u around 0 71.0%

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

      1. sub-neg 71.0%

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

   4. Simplified 71.1%

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

   5. Taylor expanded in u around 0 64.6%

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

      1. sub-neg 64.6%

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

      2. metadata-eval 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*r* 64.6%

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

      5. metadata-eval 64.6%

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

      6. sub-neg 64.6%

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

      7. expm1-def 64.6%

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

   7. Simplified 64.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \]

Alternative 9: 90.9% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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-def 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-def 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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.119999997 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in u around 0 61.9%

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

      1. sub-neg 61.9%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in u around 0 55.6%

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

   6. Taylor expanded in v around inf 55.0%

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

      1. +-commutative 55.0%

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

      2. associate-*r/ 55.0%

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

      3. metadata-eval 55.0%

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

      4. associate-*r/ 55.0%

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

      5. metadata-eval 55.0%

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

      6. unpow2 55.0%

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

   8. Simplified 55.0%

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

4. Final simplification 89.4%

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

Alternative 10: 90.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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-def 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-def 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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.119999997 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in u around 0 61.9%

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

      1. sub-neg 61.9%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in u around 0 55.6%

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

   6. Taylor expanded in v around inf 55.0%

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

      1. associate-*r/ 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

      5. unpow2 55.0%

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

   8. Simplified 55.0%

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

4. Final simplification 89.4%

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

Alternative 11: 90.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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-def 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-def 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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.119999997 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in u around 0 61.9%

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

      1. sub-neg 61.9%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in u around 0 55.6%

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

      1. sub-neg 55.6%

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

      2. metadata-eval 55.6%

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

      3. *-commutative 55.6%

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

      4. associate-*r* 55.6%

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

      5. metadata-eval 55.6%

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

      6. sub-neg 55.6%

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

      7. expm1-def 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in v around inf 55.0%

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

      1. associate-*r/ 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

      5. unpow2 55.0%

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

   10. Simplified 55.0%

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

4. Final simplification 89.4%

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

Alternative 12: 90.9% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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-def 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-def 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. Taylor expanded in v around 0 100.0%

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

   5. Taylor expanded in v around 0 92.2%

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

   if 0.119999997 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in u around 0 61.9%

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

      1. sub-neg 61.9%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in u around 0 55.6%

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

   6. Taylor expanded in v around inf 55.0%

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

      1. distribute-lft-out 55.0%

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

   8. Simplified 55.0%

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

4. Final simplification 89.4%

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

Alternative 13: 5.6% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in u around 0 5.5%

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

3. Final simplification 5.5%

   \[\leadsto -1 \]

Alternative 14: 87.5% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 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-def 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. Taylor expanded in v around 0 99.5%

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

5. Taylor expanded in v around 0 86.0%

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

6. Final simplification 86.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023182 
(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))))))))