UniformSampleCone, z

Percentage Accurate: 99.9% → 99.9%

Time: 1.7s

Alternatives: 4

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
  (+ (- 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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (+ (- 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, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (fma (- maxCos 1.0) ux 1.0))

C

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

Julia

function code(ux, uy, maxCos)
	return fma(Float32(maxCos - Float32(1.0)), ux, Float32(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. +-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. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift--.f32 N/A

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

   5. associate-+l- N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

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

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

   9. distribute-rgt1-in N/A

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

   10. +-commutative N/A

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

   11. sub-flip N/A

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

   12. sub-negate-rev N/A

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

   13. lift--.f32 N/A

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

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

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f32 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (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: 98.0% accurate, 2.6× speedup

Math

\[1 - ux \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (- 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 98.0%

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

4. Applied rewrites 98.0%

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

Alternative 4: 71.7% accurate, 9.2× speedup

Math

\[1 \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  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 98.0%

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 71.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025323 
(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)))