HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.8s

Alternatives: 12

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, 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 99.2%

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

3. Final simplification 99.2%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ \mathbf{if}\;v \leq 1:\\ \;\;\;\;1 + v \cdot \log \left(u + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{t\_0} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v))))
   (if (<= v 1.0)
     (+ 1.0 (* v (log (+ u t_0))))
     (+ (* u (* v (+ (/ 1.0 t_0) -1.0))) -1.0))))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	float tmp;
	if (v <= 1.0f) {
		tmp = 1.0f + (v * logf((u + t_0)));
	} else {
		tmp = (u * (v * ((1.0f / t_0) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = exp(((-2.0e0) / v))
    if (v <= 1.0e0) then
        tmp = 1.0e0 + (v * log((u + t_0)))
    else
        tmp = (u * (v * ((1.0e0 / t_0) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	tmp = Float32(0.0)
	if (v <= Float32(1.0))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + t_0))));
	else
		tmp = Float32(Float32(u * Float32(v * Float32(Float32(Float32(1.0) / t_0) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = exp((single(-2.0) / v));
	tmp = single(0.0);
	if (v <= single(1.0))
		tmp = single(1.0) + (v * log((u + t_0)));
	else
		tmp = (u * (v * ((single(1.0) / t_0) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.0%

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

   if 1 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.6%

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

      3. +-commutative 91.6%

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

      4. fma-define 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in u around 0 79.3%

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

4. Final simplification 97.1%

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

Alternative 3: 97.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 1

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. log-rec 98.6%

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

      4. remove-double-neg 98.6%

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

   7. Simplified 98.6%

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

   if 1 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.6%

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

      3. +-commutative 91.6%

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

      4. fma-define 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in u around 0 79.3%

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

4. Final simplification 96.8%

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

Alternative 4: 97.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 1

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. log-rec 98.6%

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

      4. remove-double-neg 98.6%

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

   7. Simplified 98.6%

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

   if 1 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.6%

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

      3. +-commutative 91.6%

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

      4. fma-define 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in u around 0 79.3%

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

      1. sub-neg 79.3%

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

      2. rec-exp 79.3%

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

      3. metadata-eval 79.3%

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

   7. Applied egg-rr 79.3%

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

      1. metadata-eval 79.3%

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

      2. sub-neg 79.3%

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

      3. expm1-undefine 79.3%

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

      4. distribute-neg-frac 79.3%

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

      5. metadata-eval 79.3%

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

   9. Simplified 79.3%

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

4. Final simplification 96.8%

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

Alternative 5: 97.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 1

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. log-rec 98.6%

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

      4. remove-double-neg 98.6%

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

   7. Simplified 98.6%

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

   if 1 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.6%

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

      3. +-commutative 91.6%

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

      4. fma-define 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in u around 0 79.3%

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

   6. Taylor expanded in v around -inf 78.1%

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

   7. Taylor expanded in u around 0 78.1%

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

      1. associate-/l* 78.1%

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

      2. associate-*r/ 78.1%

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

      3. metadata-eval 78.1%

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

   9. Simplified 78.1%

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

4. Final simplification 96.6%

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

Alternative 6: 90.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((u * ((1.3333333333333333f + (0.6666666666666666f / v)) / v)) - (u * -2.0f)) / v) + (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((u * ((1.3333333333333333e0 + (0.6666666666666666e0 / v)) / v)) - (u * (-2.0e0))) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(u * Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v)) - Float32(u * Float32(-2.0))) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((u * ((single(1.3333333333333333) + (single(0.6666666666666666) / v)) / v)) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in u around 0 70.6%

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

   6. Taylor expanded in v around -inf 70.3%

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

   7. Taylor expanded in u around 0 70.3%

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

      1. associate-/l* 70.3%

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

      2. associate-*r/ 70.3%

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

      3. metadata-eval 70.3%

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

   9. Simplified 70.3%

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

4. Final simplification 88.7%

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

Alternative 7: 90.7% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    -1.0
    (* u (* v (/ (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f - (u * (v * ((-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / v)) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) - (u * (v * (((-2.0e0) + (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(v * Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) - (u * (v * ((single(-2.0) + ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in u around 0 70.6%

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

   6. Taylor expanded in v around -inf 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-neg-frac2 69.7%

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

      3. sub-neg 69.7%

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

      4. associate-*r/ 69.7%

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

      5. distribute-lft-in 69.7%

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

      6. metadata-eval 69.7%

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

      7. neg-mul-1 69.7%

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

      8. associate-*r/ 69.7%

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

      9. metadata-eval 69.7%

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

      10. distribute-neg-frac 69.7%

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

      11. metadata-eval 69.7%

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

      12. metadata-eval 69.7%

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

   8. Simplified 69.7%

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

4. Final simplification 88.6%

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

Alternative 8: 90.4% accurate, 11.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ 1.0 (- (* u (- 2.0 (* 2.0 (/ -1.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f - (2.0f * (-1.0f / v)))) - 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 - (2.0e0 * ((-1.0e0) / v)))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) - Float32(Float32(2.0) * Float32(Float32(-1.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) - (single(2.0) * (single(-1.0) / v)))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < 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 v around inf 68.8%

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

      1. fma-define 68.8%

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

      2. associate-*r/ 68.8%

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

      3. *-commutative 68.8%

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

      4. associate-/l* 68.8%

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

      5. *-commutative 68.8%

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

      6. unpow2 68.8%

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

      7. associate-*l* 68.8%

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

      8. *-commutative 68.8%

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

      9. distribute-lft-out 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in u around 0 64.7%

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

4. Final simplification 88.1%

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

Alternative 9: 90.4% accurate, 15.2× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in u around 0 70.6%

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

   6. Taylor expanded in v around inf 64.6%

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

      1. sub-neg 64.6%

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

      2. distribute-lft-out 64.6%

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

      3. metadata-eval 64.6%

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

   8. Simplified 64.6%

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

4. Final simplification 88.1%

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

Alternative 10: 89.9% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in u around 0 70.6%

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

   6. Taylor expanded in v around inf 56.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.2% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.20000000298023224) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(-1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 91.1%

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

   if 0.200000003 < v

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. fma-define 92.6%

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

      3. +-commutative 92.6%

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

      4. fma-define 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in u around 0 45.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 5.9% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-define 99.1%

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

   3. +-commutative 99.1%

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

   4. fma-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 7.9%

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

6. Final simplification 7.9%

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

Reproduce

herbie shell --seed 2024069 
(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))))))))