UniformSampleCone, z

Percentage Accurate: 99.9% → 99.9%

Time: 1.6s

Alternatives: 5

Speedup: 1.1×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(1 - ux\right) + ux \cdot maxCos \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(1.0) - ux) + (ux * maxCos);
end

Initial Program: 99.9% accurate, 1.0× speedup

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(1 - ux\right) + ux \cdot maxCos \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(1.0) - ux) + (ux * maxCos);
end

Alternative 1: 99.9% accurate, 1.0× speedup

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(maxCos \cdot ux - ux\right) - -1 \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (- (- (* maxCos ux) ux) -1.0))

C

float code(float ux, float uy, float maxCos) {
	return ((maxCos * ux) - ux) - -1.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ((maxcos * ux) - ux) - (-1.0e0)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(maxCos * ux) - ux) - Float32(-1.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ((maxCos * ux) - ux) - single(-1.0);
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

      \[\leadsto \color{blue}{1 - \left(ux - ux \cdot maxCos\right)} \]

   4. sub-negate-rev N/A

      \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(\left(ux \cdot maxCos - ux\right)\right)\right)} \]

   5. add-flip-rev N/A

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

   6. +-commutative N/A

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

   7. add-flip-rev N/A

      \[\leadsto \color{blue}{\left(ux \cdot maxCos - ux\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(ux \cdot maxCos - ux\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

   9. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux \cdot maxCos\right)\right)\right)\right)} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{ux \cdot maxCos}\right)\right)\right)\right) - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

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

       \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos}\right)\right) - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   12. lower--.f32 N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)\right) - ux\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

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

       \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot maxCos\right)\right)}\right)\right) - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   14. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{ux \cdot maxCos}\right)\right)\right)\right) - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   15. remove-double-neg N/A

       \[\leadsto \left(\color{blue}{ux \cdot maxCos} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   16. lift-*.f32 N/A

       \[\leadsto \left(\color{blue}{ux \cdot maxCos} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(\color{blue}{maxCos \cdot ux} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \left(\color{blue}{maxCos \cdot ux} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   19. metadata-eval 99.9%

       \[\leadsto \left(maxCos \cdot ux - ux\right) - \color{blue}{-1} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(maxCos \cdot ux - ux\right) - -1} \]
4. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(ux, maxCos, 1 - ux\right) \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma ux maxCos (- 1.0 ux)))

C

float code(float ux, float uy, float maxCos) {
	return fmaf(ux, maxCos, (1.0f - ux));
}

Julia

function code(ux, uy, maxCos)
	return fma(ux, maxCos, Float32(Float32(1.0) - ux))
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]
4. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(maxCos, ux, 1\right) - ux \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (- (fma maxCos ux 1.0) ux))

C

float code(float ux, float uy, float maxCos) {
	return fmaf(maxCos, ux, 1.0f) - ux;
}

Julia

function code(ux, uy, maxCos)
	return Float32(fma(maxCos, ux, Float32(1.0)) - ux)
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

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

      \[\leadsto \color{blue}{\left(1 - ux\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos} \]

   4. sub-flip N/A

      \[\leadsto \color{blue}{\left(1 - ux\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)\right) + \left(1 - ux\right)} \]

   6. lift--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)\right) + \color{blue}{\left(1 - ux\right)} \]

   7. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)\right) + 1\right) - ux} \]

   8. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)\right) + 1\right) - ux} \]

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

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot maxCos\right)\right)}\right)\right) + 1\right) - ux \]

   10. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{ux \cdot maxCos}\right)\right)\right)\right) + 1\right) - ux \]

   11. remove-double-neg N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 99.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right) - ux} \]
4. Add Preprocessing

Alternative 4: 98.1% accurate, 2.6× speedup

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[1 - ux \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (- 1.0 ux))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f - ux;
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 - ux
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) - ux)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) - ux;
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{1 - ux} \]
3. Step-by-step derivation

   1. lower--.f32 98.1%

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

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{1 - ux} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 9.2× speedup

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[1 \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  1.0)

C

float code(float ux, float uy, float maxCos) {
	return 1.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(1.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{1 - ux} \]
3. Step-by-step derivation

   1. lower--.f32 98.1%

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in ux around 0

   \[\leadsto 1 \]
6. Step-by-step derivation

7. Applied rewrites 71.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025356 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))