Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.3%

Time: 4.7s

Alternatives: 12

Speedup: 2.7×

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: 61.3% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\\ \mathbf{if}\;u \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{fma}\left(u \cdot 4, s, \left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot s\right) \cdot \left(u \cdot u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{0 - t\_0 \cdot t\_0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (log (fma -4.0 u 1.0))))
   (if (<= u 0.009999999776482582)
     (fma
      (* u 4.0)
      s
      (* (* (fma (fma 64.0 u 21.333333333333332) u 8.0) s) (* u u)))
     (* s (/ (- 0.0 (* t_0 t_0)) t_0)))))

C

float code(float s, float u) {
	float t_0 = logf(fmaf(-4.0f, u, 1.0f));
	float tmp;
	if (u <= 0.009999999776482582f) {
		tmp = fmaf((u * 4.0f), s, ((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * s) * (u * u)));
	} else {
		tmp = s * ((0.0f - (t_0 * t_0)) / t_0);
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = log(fma(Float32(-4.0), u, Float32(1.0)))
	tmp = Float32(0.0)
	if (u <= Float32(0.009999999776482582))
		tmp = fma(Float32(u * Float32(4.0)), s, Float32(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * s) * Float32(u * u)));
	else
		tmp = Float32(s * Float32(Float32(Float32(0.0) - Float32(t_0 * t_0)) / t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00999999978

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{u \cdot \mathsf{fma}\left(4, s, u \cdot \mathsf{fma}\left(8, s, u \cdot \mathsf{fma}\left(21.333333333333332, s, 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 93.8

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

      13. lift-fma.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(u \cdot 4, \color{blue}{s}, \left(u \cdot u\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(64 \cdot u, s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. sum-to-mult N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(\left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right)\right) \cdot \left(u \cdot u\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right) \cdot \left(u \cdot u\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right)\right)\right) \]

   8. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \mathsf{fma}\left(8, \frac{s}{\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot s\right) \cdot u}, 1\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot s\right) \cdot u\right) \cdot u\right) \cdot u\right)\right) \]

   10. Applied rewrites 93.8%

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

   if 0.00999999978 < u

   1. Initial program 61.3%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f32 N/A

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

      8. metadata-eval 61.3

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

   3. Applied rewrites 61.3%

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

   5. Applied rewrites 59.8%

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

Alternative 2: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if u < 0.00999999978

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{u \cdot \mathsf{fma}\left(4, s, u \cdot \mathsf{fma}\left(8, s, u \cdot \mathsf{fma}\left(21.333333333333332, s, 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 93.8

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

      13. lift-fma.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(u \cdot 4, \color{blue}{s}, \left(u \cdot u\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(64 \cdot u, s, 21.333333333333332 \cdot s\right), u, 8 \cdot s\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. sum-to-mult N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(\left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right)\right) \cdot \left(u \cdot u\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right) \cdot \left(u \cdot u\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \left(1 + \frac{8 \cdot s}{\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right)\right)\right) \]

   8. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(u \cdot 4, s, \mathsf{fma}\left(8, \frac{s}{\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot s\right) \cdot u}, 1\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot s\right) \cdot u\right) \cdot u\right) \cdot u\right)\right) \]

   10. Applied rewrites 93.8%

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

   if 0.00999999978 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.003800000064074993:\\ \;\;\;\;\mathsf{fma}\left(u \cdot 4, s, \left(u \cdot u\right) \cdot \left(s \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.003800000064074993)
   (fma (* u 4.0) s (* (* u u) (* s (+ 8.0 (* u 21.333333333333332)))))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.003800000064074993f) {
		tmp = fmaf((u * 4.0f), s, ((u * u) * (s * (8.0f + (u * 21.333333333333332f)))));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.003800000064074993))
		tmp = fma(Float32(u * Float32(4.0)), s, Float32(Float32(u * u) * Float32(s * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332))))));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00380000006

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{u \cdot \mathsf{fma}\left(4, s, u \cdot \mathsf{fma}\left(8, s, u \cdot \mathsf{fma}\left(21.333333333333332, s, 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 93.8

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

      13. lift-fma.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 93.8

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 91.6%

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

   if 0.00380000006 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if u < 0.00999999978

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{u \cdot \mathsf{fma}\left(4, s, u \cdot \mathsf{fma}\left(8, s, u \cdot \mathsf{fma}\left(21.333333333333332, s, 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-+r+ N/A

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

      9. sum-to-mult N/A

         \[\leadsto u \cdot \left(\left(1 + \frac{8 \cdot \left(s \cdot u\right)}{4 \cdot s}\right) \cdot \left(4 \cdot s\right) + \color{blue}{\left(u \cdot \mathsf{fma}\left(\frac{64}{3}, s, 64 \cdot \left(s \cdot u\right)\right)\right)} \cdot u\right) \]

      10. *-commutative N/A

          \[\leadsto u \cdot \left(\left(1 + \frac{8 \cdot \left(s \cdot u\right)}{4 \cdot s}\right) \cdot \left(s \cdot 4\right) + \left(u \cdot \color{blue}{\mathsf{fma}\left(\frac{64}{3}, s, 64 \cdot \left(s \cdot u\right)\right)}\right) \cdot u\right) \]

      11. associate-*r* N/A

          \[\leadsto u \cdot \left(\left(\left(1 + \frac{8 \cdot \left(s \cdot u\right)}{4 \cdot s}\right) \cdot s\right) \cdot 4 + \color{blue}{\left(u \cdot \mathsf{fma}\left(\frac{64}{3}, s, 64 \cdot \left(s \cdot u\right)\right)\right)} \cdot u\right) \]

      12. lower-fma.f32 N/A

          \[\leadsto u \cdot \mathsf{fma}\left(\left(1 + \frac{8 \cdot \left(s \cdot u\right)}{4 \cdot s}\right) \cdot s, \color{blue}{4}, \left(u \cdot \mathsf{fma}\left(\frac{64}{3}, s, 64 \cdot \left(s \cdot u\right)\right)\right) \cdot u\right) \]

   6. Applied rewrites 93.0%

      \[\leadsto u \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{u \cdot s}{s}, 2, 1\right) \cdot s, \color{blue}{4}, \left(\mathsf{fma}\left(64 \cdot u, s, 21.333333333333332 \cdot s\right) \cdot u\right) \cdot u\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto u \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{u \cdot s}{s}, 2, 1\right) \cdot s, 4, \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right) \cdot u\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{u \cdot s}{s}, 2, 1\right) \cdot s, 4, \left(\mathsf{fma}\left(64 \cdot u, s, \frac{64}{3} \cdot s\right) \cdot u\right) \cdot u\right) \cdot \color{blue}{u} \]

      3. lower-*.f32 93.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{u \cdot s}{s}, 2, 1\right) \cdot s, 4, \left(\mathsf{fma}\left(64 \cdot u, s, 21.333333333333332 \cdot s\right) \cdot u\right) \cdot u\right) \cdot \color{blue}{u} \]

   8. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \mathsf{fma}\left(u, 2, 1\right), s, \left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot s\right) \cdot u\right) \cdot u\right) \cdot u} \]

   10. Applied rewrites 93.1%

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

   if 0.00999999978 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.003800000064074993:\\ \;\;\;\;s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + 21.333333333333332 \cdot u\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.003800000064074993)
   (* s (* u (+ 4.0 (* u (+ 8.0 (* 21.333333333333332 u))))))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.003800000064074993f) {
		tmp = s * (u * (4.0f + (u * (8.0f + (21.333333333333332f * u)))));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.003800000064074993))
		tmp = Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) + Float32(Float32(21.333333333333332) * u))))));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00380000006

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 90.9

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

   4. Applied rewrites 90.9%

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

   if 0.00380000006 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.003800000064074993:\\ \;\;\;\;u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.003800000064074993)
   (* u (* s (+ 4.0 (* u (+ 8.0 (* u 21.333333333333332))))))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.003800000064074993f) {
		tmp = u * (s * (4.0f + (u * (8.0f + (u * 21.333333333333332f)))));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.003800000064074993))
		tmp = Float32(u * Float32(s * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332)))))));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00380000006

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.1

         \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + 64 \cdot u\right)\right)\right)\right) \]

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 90.9%

       \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \]

   if 0.00380000006 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.0008999999845400453:\\ \;\;\;\;\mathsf{fma}\left(u \cdot 4, s, \left(\left(u \cdot s\right) \cdot 8\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.0008999999845400453)
   (fma (* u 4.0) s (* (* (* u s) 8.0) u))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.0008999999845400453f) {
		tmp = fmaf((u * 4.0f), s, (((u * s) * 8.0f) * u));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.0008999999845400453))
		tmp = fma(Float32(u * Float32(4.0)), s, Float32(Float32(Float32(u * s) * Float32(8.0)) * u));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 8.99999985e-4

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 87.1

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 87.1

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 87.1

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

   6. Applied rewrites 87.1%

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

   if 8.99999985e-4 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.0008999999845400453:\\ \;\;\;\;u \cdot \mathsf{fma}\left(4, s, 8 \cdot \left(s \cdot u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.0008999999845400453)
   (* u (fma 4.0 s (* 8.0 (* s u))))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.0008999999845400453f) {
		tmp = u * fmaf(4.0f, s, (8.0f * (s * u)));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.0008999999845400453))
		tmp = Float32(u * fma(Float32(4.0), s, Float32(Float32(8.0) * Float32(s * u))));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 8.99999985e-4

   1. Initial program 61.3%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 86.8

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

   4. Applied rewrites 86.8%

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

   if 8.99999985e-4 < u

   1. Initial program 61.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 61.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 63.9

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

      9. lift--.f32 N/A

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

      10. sub-flip N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-fma.f32 N/A

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

      15. metadata-eval 63.9

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

   3. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.3%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 86.8

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

4. Applied rewrites 86.8%

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

Alternative 10: 86.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.3%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 86.8

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

4. Applied rewrites 86.8%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 86.6

      \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right)\right) \]

7. Applied rewrites 86.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 86.6

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

9. Applied rewrites 86.6%

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

Alternative 11: 73.8% accurate, 2.7× 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 61.3%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 73.8

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

4. Applied rewrites 73.8%

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

Alternative 12: 73.6% accurate, 2.7× 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 61.3%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 73.6

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

4. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025148 
(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))))))