HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 100.0%

Time: 19.5s

Alternatives: 15

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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	double tmp = 1.0 + (((double) v) * log((((double) u) + ((1.0 - ((double) u)) * exp((-2.0 / ((double) v)))))));
	return (float) tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(8) :: tmp
    tmp = 1.0d0 + (real(v, 8) * log((real(u, 8) + ((1.0d0 - real(u, 8)) * exp(((-2.0d0) / real(v, 8)))))))
    code = real(tmp, 4)
end function

Julia

function code(u, v)
	tmp = Float64(1.0 + Float64(Float64(v) * log(Float64(Float64(u) + Float64(Float64(1.0 - Float64(u)) * exp(Float64(-2.0 / Float64(v))))))))
	return Float32(tmp)
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. rewrite-binary32/binary64 100.0%

      \[\leadsto \color{blue}{\langle \color{blue}{1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)} \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 100.0%

   \[\leadsto \color{blue}{\langle \color{blue}{1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)} \rangle_{\text{binary64}}} \]

4. Final simplification 100.0%

   \[\leadsto \langle 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \rangle_{\text{binary64}} \]

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ 1 + \langle \left( v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (cast
   (! :precision binary64 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	double tmp = ((double) v) * log((((double) u) + ((1.0 - ((double) u)) * exp((-2.0 / ((double) v))))));
	return 1.0f + ((float) tmp);
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(8) :: tmp
    tmp = real(v, 8) * log((real(u, 8) + ((1.0d0 - real(u, 8)) * exp(((-2.0d0) / real(v, 8))))))
    code = 1.0e0 + real(tmp, 4)
end function

Julia

function code(u, v)
	tmp = Float64(Float64(v) * log(Float64(Float64(u) + Float64(Float64(1.0 - Float64(u)) * exp(Float64(-2.0 / Float64(v)))))))
	return Float32(Float32(1.0) + Float32(tmp))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. rewrite-binary32/binary64 99.9%

      \[\leadsto \color{blue}{1 + \langle v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 99.9%

   \[\leadsto 1 + \color{blue}{\langle \color{blue}{v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)} \rangle_{\text{binary64}}} \]

4. Final simplification 99.9%

   \[\leadsto 1 + \langle v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \rangle_{\text{binary64}} \]

Alternative 3: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \langle \left( \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}}, 1\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (fma
  v
  (cast (! :precision binary64 (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))))
  1.0))

C

float code(float u, float v) {
	double tmp = log((((double) u) + ((1.0 - ((double) u)) * exp((-2.0 / ((double) v))))));
	return fmaf(v, ((float) tmp), 1.0f);
}

Julia

function code(u, v)
	tmp = log(Float64(Float64(u) + Float64(Float64(1.0 - Float64(u)) * exp(Float64(-2.0 / Float64(v))))))
	return fma(v, Float32(tmp), Float32(1.0))
end

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-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. Step-by-step derivation

   1. fma-udef 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. Applied egg-rr 99.5%

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

   1. rewrite-binary32/binary64 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \langle \log \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u\right) \rangle_{\text{binary64}}, 1\right)} \]

7. Applied rewrite-once 99.9%

   \[\leadsto \mathsf{fma}\left(v, \color{blue}{\langle \color{blue}{\log \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u\right)} \rangle_{\text{binary64}}}, 1\right) \]

8. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(v, \langle \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \rangle_{\text{binary64}}, 1\right) \]

Alternative 4: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \langle \left( \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (cast (! :precision binary64 (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	double tmp = log((((double) u) + ((1.0 - ((double) u)) * exp((-2.0 / ((double) v))))));
	return 1.0f + (v * ((float) tmp));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(8) :: tmp
    tmp = log((real(u, 8) + ((1.0d0 - real(u, 8)) * exp(((-2.0d0) / real(v, 8))))))
    code = 1.0e0 + (v * real(tmp, 4))
end function

Julia

function code(u, v)
	tmp = log(Float64(Float64(u) + Float64(Float64(1.0 - Float64(u)) * exp(Float64(-2.0 / Float64(v))))))
	return Float32(Float32(1.0) + Float32(v * Float32(tmp)))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. rewrite-binary32/binary64 99.8%

      \[\leadsto \color{blue}{1 + v \cdot \langle \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 99.8%

   \[\leadsto 1 + v \cdot \color{blue}{\langle \color{blue}{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)} \rangle_{\text{binary64}}} \]

4. Final simplification 99.8%

   \[\leadsto 1 + v \cdot \langle \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \rangle_{\text{binary64}} \]

Alternative 5: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-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. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-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. Step-by-step derivation

   1. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Final simplification 99.5%

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

Alternative 8: 91.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot -24 + 8\right)}{v \cdot v} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + \left(1 - u\right) \cdot 4}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* -2.0 (- 1.0 u))
     (+
      (* 0.16666666666666666 (/ (* u (+ (* u -24.0) 8.0)) (* v v)))
      (* 0.5 (/ (+ (* -4.0 (pow (- 1.0 u) 2.0)) (* (- 1.0 u) 4.0)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((-2.0f * (1.0f - u)) + ((0.16666666666666666f * ((u * ((u * -24.0f) + 8.0f)) / (v * v))) + (0.5f * (((-4.0f * powf((1.0f - u), 2.0f)) + ((1.0f - u) * 4.0f)) / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((-2.0e0) * (1.0e0 - u)) + ((0.16666666666666666e0 * ((u * ((u * (-24.0e0)) + 8.0e0)) / (v * v))) + (0.5e0 * ((((-4.0e0) * ((1.0e0 - u) ** 2.0e0)) + ((1.0e0 - u) * 4.0e0)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(-2.0) * Float32(Float32(1.0) - u)) + Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(u * Float32(-24.0)) + Float32(8.0))) / Float32(v * v))) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(-2.0) * (single(1.0) - u)) + ((single(0.16666666666666666) * ((u * ((u * single(-24.0)) + single(8.0))) / (v * v))) + (single(0.5) * (((single(-4.0) * ((single(1.0) - u) ^ single(2.0))) + ((single(1.0) - u) * single(4.0))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. fma-udef 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. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.3%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   2. Taylor expanded in v around inf 70.4%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)} \]

   3. Taylor expanded in u around 0 66.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{-24 \cdot {u}^{2} + 8 \cdot u}}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
   4. Step-by-step derivation

      1. fma-def 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{\mathsf{fma}\left(-24, {u}^{2}, 8 \cdot u\right)}}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      2. unpow2 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-24, \color{blue}{u \cdot u}, 8 \cdot u\right)}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      3. *-commutative 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-24, u \cdot u, \color{blue}{u \cdot 8}\right)}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

   5. Simplified 66.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{\mathsf{fma}\left(-24, u \cdot u, u \cdot 8\right)}}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

   6. Taylor expanded in v around 0 66.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(\color{blue}{0.16666666666666666 \cdot \frac{-24 \cdot {u}^{2} + 8 \cdot u}{{v}^{2}}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{-24 \cdot \color{blue}{\left(u \cdot u\right)} + 8 \cdot u}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      2. associate-*l* 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{\left(-24 \cdot u\right) \cdot u} + 8 \cdot u}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      3. distribute-rgt-out 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{u \cdot \left(-24 \cdot u + 8\right)}}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      4. *-commutative 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{u \cdot \left(\color{blue}{u \cdot -24} + 8\right)}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

      5. unpow2 66.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot -24 + 8\right)}{\color{blue}{v \cdot v}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]

   8. Simplified 66.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(\color{blue}{0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot -24 + 8\right)}{v \cdot v}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{u \cdot \left(u \cdot -24 + 8\right)}{v \cdot v} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + \left(1 - u\right) \cdot 4}{v}\right)\right)\\ \end{array} \]

Alternative 9: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ (* (expm1 (/ 2.0 v)) (* v u)) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (expm1f((2.0f / v)) * (v * u)) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(expm1(Float32(Float32(2.0) / v)) * Float32(v * u)) + Float32(-1.0));
	end
	return 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. 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. Step-by-step derivation

      1. fma-udef 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. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 93.9%

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

   if 0.200000003 < v

   1. Initial program 92.4%

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

   2. Taylor expanded in u around 0 69.4%

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

      1. associate-*r* 69.4%

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

      2. sub-neg 69.4%

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

      3. distribute-lft-in 69.4%

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

      4. rec-exp 69.4%

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

      5. distribute-neg-frac 69.4%

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

      6. metadata-eval 69.4%

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

      7. metadata-eval 69.4%

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

   4. Applied egg-rr 69.4%

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

      1. metadata-eval 69.4%

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

      2. distribute-neg-frac 69.4%

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

      3. rec-exp 69.4%

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

      4. distribute-lft-in 69.4%

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

      5. metadata-eval 69.4%

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

      6. sub-neg 69.4%

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

      7. *-commutative 69.4%

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

      8. rec-exp 69.4%

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

      9. distribute-neg-frac 69.4%

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

      10. metadata-eval 69.4%

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

      11. metadata-eval 69.4%

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

      12. associate-*r/ 69.4%

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

      13. expm1-def 69.4%

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

      14. associate-*r/ 69.4%

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

      15. metadata-eval 69.4%

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

      16. *-commutative 69.4%

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

   6. Simplified 69.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right) + -1\\ \end{array} \]

Alternative 10: 90.6% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((-2.0e0) * (1.0e0 - u)) + (0.5e0 * ((((1.0e0 - u) * 4.0e0) + ((-4.0e0) * ((1.0e0 - u) * (1.0e0 - u)))) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(-2.0) * (single(1.0) - u)) + (single(0.5) * ((((single(1.0) - u) * single(4.0)) + (single(-4.0) * ((single(1.0) - u) * (single(1.0) - u)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. fma-udef 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. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.3%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   2. Taylor expanded in v around inf 64.0%

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

      1. unpow2 64.0%

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

   4. Applied egg-rr 64.0%

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

4. Final simplification 92.0%

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

Alternative 11: 90.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-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. Step-by-step derivation

      1. fma-udef 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. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.3%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   2. Taylor expanded in u around 0 65.9%

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

   3. Taylor expanded in v around inf 60.8%

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

      1. distribute-lft-out 60.8%

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

   5. Simplified 60.8%

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

4. Final simplification 91.8%

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

Alternative 12: 89.8% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. fma-udef 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. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.3%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   2. Taylor expanded in v around inf 50.8%

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

4. Final simplification 91.0%

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

Alternative 13: 89.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. fma-udef 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. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.3%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   2. Taylor expanded in v around inf 50.8%

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

   3. Taylor expanded in u around 0 50.7%

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

4. Final simplification 91.0%

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

Alternative 14: 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 99.5%

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

2. Taylor expanded in u around 0 6.0%

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

3. Final simplification 6.0%

   \[\leadsto -1 \]

Alternative 15: 86.8% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-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. Step-by-step derivation

   1. fma-udef 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. Applied egg-rr 99.5%

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

6. Taylor expanded in v around 0 87.8%

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

7. Final simplification 87.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(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))))))))