UniformSampleCone, z

Percentage Accurate: 99.9% → 99.9%

Time: 4.7s

Alternatives: 5

Speedup: 1.0×

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

\[\begin{array}{l} \\ \left(1 - ux\right) + ux \cdot maxCos \end{array} \]

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)
    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

\[\begin{array}{l} \\ \left(1 - ux\right) + ux \cdot maxCos \end{array} \]

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)
    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, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(ux + \left(ux \cdot -2 + ux \cdot maxCos\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (+ 1.0 (+ ux (+ (* ux -2.0) (* ux maxCos)))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f + (ux + ((ux * -2.0f) + (ux * maxCos)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 + (ux + ((ux * (-2.0e0)) + (ux * maxcos)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) + Float32(ux + Float32(Float32(ux * Float32(-2.0)) + Float32(ux * maxCos))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) + (ux + ((ux * single(-2.0)) + (ux * maxCos)));
end

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 100.0%

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

   2. flip-- 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-un-lft-identity 98.9%

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

   5. *-commutative 98.9%

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

   6. distribute-rgt-out-- 98.9%

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

   7. *-un-lft-identity 98.9%

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

   8. *-commutative 98.9%

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

   9. distribute-rgt-out-- 98.9%

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

   10. *-un-lft-identity 98.9%

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

   11. *-commutative 98.9%

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

   12. distribute-rgt-out-- 98.9%

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

3. Applied egg-rr 98.9%

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

4. Taylor expanded in maxCos around inf 98.5%

   \[\leadsto \color{blue}{\left(-2 \cdot ux + maxCos \cdot ux\right) - -1 \cdot \left(1 + ux\right)} \]
5. Step-by-step derivation

   1. distribute-rgt-out 98.5%

      \[\leadsto \color{blue}{ux \cdot \left(-2 + maxCos\right)} - -1 \cdot \left(1 + ux\right) \]

   2. distribute-lft-in 98.5%

      \[\leadsto ux \cdot \left(-2 + maxCos\right) - \color{blue}{\left(-1 \cdot 1 + -1 \cdot ux\right)} \]

   3. metadata-eval 98.5%

      \[\leadsto ux \cdot \left(-2 + maxCos\right) - \left(\color{blue}{-1} + -1 \cdot ux\right) \]

   4. neg-mul-1 98.5%

      \[\leadsto ux \cdot \left(-2 + maxCos\right) - \left(-1 + \color{blue}{\left(-ux\right)}\right) \]

6. Simplified 98.5%

   \[\leadsto \color{blue}{ux \cdot \left(-2 + maxCos\right) - \left(-1 + \left(-ux\right)\right)} \]

7. Taylor expanded in maxCos around 0 100.0%

   \[\leadsto \color{blue}{1 + \left(ux + \left(-2 \cdot ux + maxCos \cdot ux\right)\right)} \]

8. Final simplification 100.0%

   \[\leadsto 1 + \left(ux + \left(ux \cdot -2 + ux \cdot maxCos\right)\right) \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - ux\right) + ux \cdot maxCos \end{array} \]

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)
    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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - ux \cdot \left(1 - maxCos\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. *-commutative 100.0%

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

   4. distribute-rgt-out-- 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 98.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 1 - ux \end{array} \]

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)
    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 98.0%

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

3. Final simplification 98.0%

   \[\leadsto 1 - ux \]

Alternative 5: 71.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

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)
    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 ux around 0 72.1%

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

3. Final simplification 72.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(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)))