Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.8%

Time: 5.2s

Alternatives: 6

Speedup: N/A×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u \cdot 4\right)\\ \mathbf{if}\;u \leq 0.003000000026077032:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot s, u, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot s\right) \cdot \frac{\mathsf{log1p}\left(\left(-1 \cdot u\right) \cdot 4\right) \cdot t\_0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (log (- 1.0 (* u 4.0)))))
   (if (<= u 0.003000000026077032)
     (fma
      (* 4.0 s)
      u
      (*
       (*
        (- (fma (* u s) 21.333333333333332 (* (* (* u s) 64.0) u)) (* -8.0 s))
        u)
       u))
     (* (* -1.0 s) (/ (* (log1p (* (* -1.0 u) 4.0)) t_0) t_0)))))

C

float code(float s, float u) {
	float t_0 = logf((1.0f - (u * 4.0f)));
	float tmp;
	if (u <= 0.003000000026077032f) {
		tmp = fmaf((4.0f * s), u, (((fmaf((u * s), 21.333333333333332f, (((u * s) * 64.0f) * u)) - (-8.0f * s)) * u) * u));
	} else {
		tmp = (-1.0f * s) * ((log1pf(((-1.0f * u) * 4.0f)) * t_0) / t_0);
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = log(Float32(Float32(1.0) - Float32(u * Float32(4.0))))
	tmp = Float32(0.0)
	if (u <= Float32(0.003000000026077032))
		tmp = fma(Float32(Float32(4.0) * s), u, Float32(Float32(Float32(fma(Float32(u * s), Float32(21.333333333333332), Float32(Float32(Float32(u * s) * Float32(64.0)) * u)) - Float32(Float32(-8.0) * s)) * u) * u));
	else
		tmp = Float32(Float32(Float32(-1.0) * s) * Float32(Float32(log1p(Float32(Float32(Float32(-1.0) * u) * Float32(4.0))) * t_0) / t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00300000003

   1. Initial program 50.0%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u} \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(4 \cdot s, \color{blue}{u}, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right) \]

   if 0.00300000003 < u

   1. Initial program 92.4%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto s \cdot \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right)} \]

      2. lift-/.f32 N/A

         \[\leadsto s \cdot \log \color{blue}{\left(\frac{1}{1 - 4 \cdot u}\right)} \]

      3. lift--.f32 N/A

         \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{1 - 4 \cdot u}}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{4 \cdot u}}\right) \]

      5. log-div N/A

         \[\leadsto s \cdot \color{blue}{\left(\log 1 - \log \left(1 - 4 \cdot u\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto s \cdot \left(\color{blue}{0} - \log \left(1 - 4 \cdot u\right)\right) \]

      7. flip-- N/A

         \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}{0 + \log \left(1 - 4 \cdot u\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}{0 + \log \left(1 - 4 \cdot u\right)}} \]

   4. Applied rewrites 94.1%

      \[\leadsto s \cdot \color{blue}{\frac{0 - \log \left(1 - u \cdot 4\right) \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto s \cdot \frac{0 - \color{blue}{\log \left(1 - u \cdot 4\right)} \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

      2. lift--.f32 N/A

         \[\leadsto s \cdot \frac{0 - \log \color{blue}{\left(1 - u \cdot 4\right)} \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

      3. lift-*.f32 N/A

         \[\leadsto s \cdot \frac{0 - \log \left(1 - \color{blue}{u \cdot 4}\right) \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto s \cdot \frac{0 - \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u\right)\right) \cdot 4\right)} \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

      5. lower-log1p.f32 N/A

         \[\leadsto s \cdot \frac{0 - \color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u\right)\right) \cdot 4\right)} \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{0 - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u\right)\right) \cdot 4}\right) \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

      7. lower-neg.f32 98.8

         \[\leadsto s \cdot \frac{0 - \mathsf{log1p}\left(\color{blue}{\left(-u\right)} \cdot 4\right) \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]

   6. Applied rewrites 98.8%

      \[\leadsto s \cdot \frac{0 - \color{blue}{\mathsf{log1p}\left(\left(-u\right) \cdot 4\right)} \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.003000000026077032:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot s, u, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot s\right) \cdot \frac{\mathsf{log1p}\left(\left(-1 \cdot u\right) \cdot 4\right) \cdot \log \left(1 - u \cdot 4\right)}{\log \left(1 - u \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u \cdot 4\right)\\ \mathbf{if}\;u \leq 0.00800000037997961:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot s, u, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot s\right) \cdot \frac{t\_0 \cdot t\_0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (log (- 1.0 (* u 4.0)))))
   (if (<= u 0.00800000037997961)
     (fma
      (* 4.0 s)
      u
      (*
       (*
        (- (fma (* u s) 21.333333333333332 (* (* (* u s) 64.0) u)) (* -8.0 s))
        u)
       u))
     (* (* -1.0 s) (/ (* t_0 t_0) t_0)))))

C

float code(float s, float u) {
	float t_0 = logf((1.0f - (u * 4.0f)));
	float tmp;
	if (u <= 0.00800000037997961f) {
		tmp = fmaf((4.0f * s), u, (((fmaf((u * s), 21.333333333333332f, (((u * s) * 64.0f) * u)) - (-8.0f * s)) * u) * u));
	} else {
		tmp = (-1.0f * s) * ((t_0 * t_0) / t_0);
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = log(Float32(Float32(1.0) - Float32(u * Float32(4.0))))
	tmp = Float32(0.0)
	if (u <= Float32(0.00800000037997961))
		tmp = fma(Float32(Float32(4.0) * s), u, Float32(Float32(Float32(fma(Float32(u * s), Float32(21.333333333333332), Float32(Float32(Float32(u * s) * Float32(64.0)) * u)) - Float32(Float32(-8.0) * s)) * u) * u));
	else
		tmp = Float32(Float32(Float32(-1.0) * s) * Float32(Float32(t_0 * t_0) / t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00800000038

   1. Initial program 53.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u} \]

   7. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(4 \cdot s, \color{blue}{u}, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right) \]

   if 0.00800000038 < u

   1. Initial program 95.0%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto s \cdot \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right)} \]

      2. lift-/.f32 N/A

         \[\leadsto s \cdot \log \color{blue}{\left(\frac{1}{1 - 4 \cdot u}\right)} \]

      3. lift--.f32 N/A

         \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{1 - 4 \cdot u}}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{4 \cdot u}}\right) \]

      5. log-div N/A

         \[\leadsto s \cdot \color{blue}{\left(\log 1 - \log \left(1 - 4 \cdot u\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto s \cdot \left(\color{blue}{0} - \log \left(1 - 4 \cdot u\right)\right) \]

      7. flip-- N/A

         \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}{0 + \log \left(1 - 4 \cdot u\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}{0 + \log \left(1 - 4 \cdot u\right)}} \]

   4. Applied rewrites 96.7%

      \[\leadsto s \cdot \color{blue}{\frac{0 - \log \left(1 - u \cdot 4\right) \cdot \log \left(1 - u \cdot 4\right)}{0 + \log \left(1 - u \cdot 4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.00800000037997961:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot s, u, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot s\right) \cdot \frac{\log \left(1 - u \cdot 4\right) \cdot \log \left(1 - u \cdot 4\right)}{\log \left(1 - u \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot s, u, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.009999999776482582)
   (fma
    (* 4.0 s)
    u
    (*
     (*
      (- (fma (* u s) 21.333333333333332 (* (* (* u s) 64.0) u)) (* -8.0 s))
      u)
     u))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.009999999776482582f) {
		tmp = fmaf((4.0f * s), u, (((fmaf((u * s), 21.333333333333332f, (((u * s) * 64.0f) * u)) - (-8.0f * s)) * u) * u));
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.009999999776482582))
		tmp = fma(Float32(Float32(4.0) * s), u, Float32(Float32(Float32(fma(Float32(u * s), Float32(21.333333333333332), Float32(Float32(Float32(u * s) * Float32(64.0)) * u)) - Float32(Float32(-8.0) * s)) * u) * u));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00999999978

   1. Initial program 53.4%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u} \]

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(4 \cdot s, \color{blue}{u}, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right) \]

   if 0.00999999978 < u

   1. Initial program 95.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(4 \cdot s, u, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma
  (* 4.0 s)
  u
  (*
   (* (- (fma (* u s) 21.333333333333332 (* (* (* u s) 64.0) u)) (* -8.0 s)) u)
   u)))

C

float code(float s, float u) {
	return fmaf((4.0f * s), u, (((fmaf((u * s), 21.333333333333332f, (((u * s) * 64.0f) * u)) - (-8.0f * s)) * u) * u));
}

Julia

function code(s, u)
	return fma(Float32(Float32(4.0) * s), u, Float32(Float32(Float32(fma(Float32(u * s), Float32(21.333333333333332), Float32(Float32(Float32(u * s) * Float32(64.0)) * u)) - Float32(Float32(-8.0) * s)) * u) * u))
end

Derivation

1. Initial program 58.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u} \]

7. Applied rewrites 95.5%

   \[\leadsto \mathsf{fma}\left(4 \cdot s, \color{blue}{u}, \left(\left(\mathsf{fma}\left(u \cdot s, 21.333333333333332, \left(\left(u \cdot s\right) \cdot 64\right) \cdot u\right) - -8 \cdot s\right) \cdot u\right) \cdot u\right) \]
8. Add Preprocessing

Alternative 5: 93.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (fma
   (fma (fma 64.0 (* u s) (* 21.333333333333332 s)) u (* 8.0 s))
   u
   (* 4.0 s))
  u))

C

float code(float s, float u) {
	return fmaf(fmaf(fmaf(64.0f, (u * s), (21.333333333333332f * s)), u, (8.0f * s)), u, (4.0f * s)) * u;
}

Julia

function code(s, u)
	return Float32(fma(fma(fma(Float32(64.0), Float32(u * s), Float32(Float32(21.333333333333332) * s)), u, Float32(Float32(8.0) * s)), u, Float32(Float32(4.0) * s)) * u)
end

Derivation

1. Initial program 58.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u} \]
6. Add Preprocessing

Alternative 6: 93.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 + u \cdot \left(21.333333333333332 - -64 \cdot u\right)\\ \frac{s \cdot \left(64 + {u}^{3} \cdot {t\_0}^{3}\right)}{\left(16 + \left(u \cdot u\right) \cdot {t\_0}^{2}\right) - 4 \cdot \left(u \cdot t\_0\right)} \cdot u \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (+ 8.0 (* u (- 21.333333333333332 (* -64.0 u))))))
   (*
    (/
     (* s (+ 64.0 (* (pow u 3.0) (pow t_0 3.0))))
     (- (+ 16.0 (* (* u u) (pow t_0 2.0))) (* 4.0 (* u t_0))))
    u)))

C

float code(float s, float u) {
	float t_0 = 8.0f + (u * (21.333333333333332f - (-64.0f * u)));
	return ((s * (64.0f + (powf(u, 3.0f) * powf(t_0, 3.0f)))) / ((16.0f + ((u * u) * powf(t_0, 2.0f))) - (4.0f * (u * t_0)))) * u;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    t_0 = 8.0e0 + (u * (21.333333333333332e0 - ((-64.0e0) * u)))
    code = ((s * (64.0e0 + ((u ** 3.0e0) * (t_0 ** 3.0e0)))) / ((16.0e0 + ((u * u) * (t_0 ** 2.0e0))) - (4.0e0 * (u * t_0)))) * u
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(8.0) + Float32(u * Float32(Float32(21.333333333333332) - Float32(Float32(-64.0) * u))))
	return Float32(Float32(Float32(s * Float32(Float32(64.0) + Float32((u ^ Float32(3.0)) * (t_0 ^ Float32(3.0))))) / Float32(Float32(Float32(16.0) + Float32(Float32(u * u) * (t_0 ^ Float32(2.0)))) - Float32(Float32(4.0) * Float32(u * t_0)))) * u)
end

MATLAB

function tmp = code(s, u)
	t_0 = single(8.0) + (u * (single(21.333333333333332) - (single(-64.0) * u)));
	tmp = ((s * (single(64.0) + ((u ^ single(3.0)) * (t_0 ^ single(3.0))))) / ((single(16.0) + ((u * u) * (t_0 ^ single(2.0)))) - (single(4.0) * (u * t_0)))) * u;
end

Derivation

1. Initial program 58.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \cdot \color{blue}{u} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u \cdot s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u} \]

6. Taylor expanded in s around 0

   \[\leadsto \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right) \cdot u \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot s\right) \cdot u \]

   2. *-commutative N/A

      \[\leadsto \left(\left(4 + \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u\right) \cdot s\right) \cdot u \]

   3. +-commutative N/A

      \[\leadsto \left(\left(4 + \left(8 + u \cdot \left(64 \cdot u + \frac{64}{3}\right)\right) \cdot u\right) \cdot s\right) \cdot u \]

   4. *-commutative N/A

      \[\leadsto \left(\left(4 + \left(8 + \left(64 \cdot u + \frac{64}{3}\right) \cdot u\right) \cdot u\right) \cdot s\right) \cdot u \]

   5. +-commutative N/A

      \[\leadsto \left(\left(4 + \left(\left(64 \cdot u + \frac{64}{3}\right) \cdot u + 8\right) \cdot u\right) \cdot s\right) \cdot u \]

   6. +-commutative N/A

      \[\leadsto \left(\left(\left(\left(64 \cdot u + \frac{64}{3}\right) \cdot u + 8\right) \cdot u + 4\right) \cdot s\right) \cdot u \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(\left(\left(64 \cdot u + \frac{64}{3}\right) \cdot u + 8\right) \cdot u + 4\right) \cdot s\right) \cdot u \]

8. Applied rewrites 94.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332 - -64 \cdot u, u, 8\right), u, 4\right) \cdot s\right) \cdot u \]

10. Applied rewrites 94.6%

    \[\leadsto \left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(21.333333333333332 - -64 \cdot u, u, 8\right)\right)}^{3}, \left(u \cdot u\right) \cdot u, 64\right)}{\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332 - -64 \cdot u, u, 8\right) \cdot u, \mathsf{fma}\left(21.333333333333332 - -64 \cdot u, u, 8\right) \cdot u, 16 - \left(\mathsf{fma}\left(21.333333333333332 - -64 \cdot u, u, 8\right) \cdot u\right) \cdot 4\right)} \cdot s\right) \cdot u \]

11. Taylor expanded in s around 0

    \[\leadsto \frac{s \cdot \left(64 + {u}^{3} \cdot {\left(8 + u \cdot \left(\frac{64}{3} - -64 \cdot u\right)\right)}^{3}\right)}{\left(16 + {u}^{2} \cdot {\left(8 + u \cdot \left(\frac{64}{3} - -64 \cdot u\right)\right)}^{2}\right) - 4 \cdot \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} - -64 \cdot u\right)\right)\right)} \cdot u \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \frac{s \cdot \left(64 + {u}^{3} \cdot {\left(8 + u \cdot \left(\frac{64}{3} - -64 \cdot u\right)\right)}^{3}\right)}{\left(16 + {u}^{2} \cdot {\left(8 + u \cdot \left(\frac{64}{3} - -64 \cdot u\right)\right)}^{2}\right) - 4 \cdot \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} - -64 \cdot u\right)\right)\right)} \cdot u \]

13. Applied rewrites 94.8%

    \[\leadsto \frac{s \cdot \left(64 + {u}^{3} \cdot {\left(8 + u \cdot \left(21.333333333333332 - -64 \cdot u\right)\right)}^{3}\right)}{\left(16 + \left(u \cdot u\right) \cdot {\left(8 + u \cdot \left(21.333333333333332 - -64 \cdot u\right)\right)}^{2}\right) - 4 \cdot \left(u \cdot \left(8 + u \cdot \left(21.333333333333332 - -64 \cdot u\right)\right)\right)} \cdot u \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))