Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 8.6s

Alternatives: 12

Speedup: 11.4×

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: 61.5% 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: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-4 \cdot u\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log1p (* -4.0 u)) (- s)))

C

float code(float s, float u) {
	return log1pf((-4.0f * u)) * -s;
}

Julia

function code(s, u)
	return Float32(log1p(Float32(Float32(-4.0) * u)) * Float32(-s))
end

Derivation

1. Initial program 60.7%

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot s} \]

5. Final simplification 99.6%

   \[\leadsto \mathsf{log1p}\left(-4 \cdot u\right) \cdot \left(-s\right) \]
6. Add Preprocessing

Alternative 2: 93.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 93.2%

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

Alternative 3: 93.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 92.9

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

5. Applied rewrites 92.9%

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

Alternative 4: 91.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot s} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 91.0

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

7. Applied rewrites 91.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4\right) \cdot u\right)} \cdot s \]
8. Step-by-step derivation

9. Applied rewrites 91.3%

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

Alternative 5: 91.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 91.2

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

5. Applied rewrites 91.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4 \cdot s\right) \cdot u} \]
6. Step-by-step derivation

7. Applied rewrites 91.2%

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

Alternative 6: 91.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 91.2

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

5. Applied rewrites 91.2%

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

Alternative 7: 91.0% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot s} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 91.0

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

7. Applied rewrites 91.0%

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

Alternative 8: 90.7% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 91.2

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

5. Applied rewrites 91.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4 \cdot s\right) \cdot u} \]
6. Step-by-step derivation

7. Applied rewrites 90.5%

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

Alternative 9: 87.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot s} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 86.9

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

7. Applied rewrites 86.9%

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

Alternative 10: 86.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 86.7

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

5. Applied rewrites 86.7%

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

Alternative 11: 74.1% accurate, 11.4× speedup

Math

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

FPCore

(FPCore (s u) :precision binary32 (* s (* 4.0 u)))

C

float code(float s, float u) {
	return s * (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 * (4.0e0 * u)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(4.0) * u))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (single(4.0) * u);
end

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
6. Add Preprocessing

Alternative 12: 73.9% accurate, 11.4× speedup

Math

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

FPCore

(FPCore (s u) :precision binary32 (* 4.0 (* s u)))

C

float code(float s, float u) {
	return 4.0f * (s * 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 = 4.0e0 * (s * u)
end function

Julia

function code(s, u)
	return Float32(Float32(4.0) * Float32(s * u))
end

MATLAB

function tmp = code(s, u)
	tmp = single(4.0) * (s * u);
end

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 86.7

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

5. Applied rewrites 86.7%

   \[\leadsto \color{blue}{\left(s \cdot \mathsf{fma}\left(8, u, 4\right)\right) \cdot u} \]
6. Step-by-step derivation

7. Applied rewrites 86.4%

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

8. Taylor expanded in u around 0

   \[\leadsto 4 \cdot \left(\color{blue}{s} \cdot u\right) \]
9. Step-by-step derivation

10. Applied rewrites 74.9%

    \[\leadsto 4 \cdot \left(\color{blue}{s} \cdot u\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(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))))))