Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 7

Speedup: 1.2×

• 21.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Python

def code(x, y, z):
	return x + ((y - x) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Python

def code(x, y, z):
	return x + ((y - x) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, z, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) z x))

C

double code(double x, double y, double z) {
	return fma((y - x), z, x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), z, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * z + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(y - x\right) \cdot z} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot z + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot z} + x \]

   4. lower-fma.f64 100.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]
5. Add Preprocessing

Alternative 2: 60.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot z\\ \mathbf{if}\;z \leq -5 \cdot 10^{+174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+83}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+99}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+226}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x) z)))
   (if (<= z -5e+174)
     t_0
     (if (<= z -1.55e+83)
       (* z y)
       (if (<= z -1.0)
         t_0
         (if (<= z 8.2e-10)
           (* 1.0 x)
           (if (<= z 3.2e+99) (* z y) (if (<= z 7e+226) t_0 (* z y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = -x * z;
	double tmp;
	if (z <= -5e+174) {
		tmp = t_0;
	} else if (z <= -1.55e+83) {
		tmp = z * y;
	} else if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 8.2e-10) {
		tmp = 1.0 * x;
	} else if (z <= 3.2e+99) {
		tmp = z * y;
	} else if (z <= 7e+226) {
		tmp = t_0;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x * z
    if (z <= (-5d+174)) then
        tmp = t_0
    else if (z <= (-1.55d+83)) then
        tmp = z * y
    else if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 8.2d-10) then
        tmp = 1.0d0 * x
    else if (z <= 3.2d+99) then
        tmp = z * y
    else if (z <= 7d+226) then
        tmp = t_0
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -x * z;
	double tmp;
	if (z <= -5e+174) {
		tmp = t_0;
	} else if (z <= -1.55e+83) {
		tmp = z * y;
	} else if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 8.2e-10) {
		tmp = 1.0 * x;
	} else if (z <= 3.2e+99) {
		tmp = z * y;
	} else if (z <= 7e+226) {
		tmp = t_0;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x * z
	tmp = 0
	if z <= -5e+174:
		tmp = t_0
	elif z <= -1.55e+83:
		tmp = z * y
	elif z <= -1.0:
		tmp = t_0
	elif z <= 8.2e-10:
		tmp = 1.0 * x
	elif z <= 3.2e+99:
		tmp = z * y
	elif z <= 7e+226:
		tmp = t_0
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) * z)
	tmp = 0.0
	if (z <= -5e+174)
		tmp = t_0;
	elseif (z <= -1.55e+83)
		tmp = Float64(z * y);
	elseif (z <= -1.0)
		tmp = t_0;
	elseif (z <= 8.2e-10)
		tmp = Float64(1.0 * x);
	elseif (z <= 3.2e+99)
		tmp = Float64(z * y);
	elseif (z <= 7e+226)
		tmp = t_0;
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x * z;
	tmp = 0.0;
	if (z <= -5e+174)
		tmp = t_0;
	elseif (z <= -1.55e+83)
		tmp = z * y;
	elseif (z <= -1.0)
		tmp = t_0;
	elseif (z <= 8.2e-10)
		tmp = 1.0 * x;
	elseif (z <= 3.2e+99)
		tmp = z * y;
	elseif (z <= 7e+226)
		tmp = t_0;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) * z), $MachinePrecision]}, If[LessEqual[z, -5e+174], t$95$0, If[LessEqual[z, -1.55e+83], N[(z * y), $MachinePrecision], If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 8.2e-10], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 3.2e+99], N[(z * y), $MachinePrecision], If[LessEqual[z, 7e+226], t$95$0, N[(z * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999997e174 or -1.54999999999999996e83 < z < -1 or 3.19999999999999999e99 < z < 6.9999999999999996e226

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

      3. lower--.f64 99.0

         \[\leadsto \color{blue}{\left(y - x\right)} \cdot z \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot x\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

      \[\leadsto \left(-x\right) \cdot z \]

   if -4.9999999999999997e174 < z < -1.54999999999999996e83 or 8.1999999999999996e-10 < z < 3.19999999999999999e99 or 6.9999999999999996e226 < z

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.3

         \[\leadsto \color{blue}{y \cdot z} \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -1 < z < 8.1999999999999996e-10

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(x \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{z \cdot x}\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot x} \]

      4. mul-1-neg N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \cdot x \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

      10. lower--.f64 78.5

          \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\left(1 - z\right) \cdot x} \]

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 77.8%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+174}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+83}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+99}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+226}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y - x\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-7}:\\ \;\;\;\;z \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- y x))))
   (if (<= z -1.0) t_0 (if (<= z 9.4e-7) (+ (* z y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y - x);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 9.4e-7) {
		tmp = (z * y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y - x)
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 9.4d-7) then
        tmp = (z * y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y - x);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 9.4e-7) {
		tmp = (z * y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y - x)
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 9.4e-7:
		tmp = (z * y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y - x))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 9.4e-7)
		tmp = Float64(Float64(z * y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y - x);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 9.4e-7)
		tmp = (z * y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 9.4e-7], N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 9.4e-7 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

      3. lower--.f64 99.3

         \[\leadsto \color{blue}{\left(y - x\right)} \cdot z \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

   if -1 < z < 9.4e-7

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 99.1

         \[\leadsto x + \color{blue}{y \cdot z} \]

   5. Applied rewrites 99.1%

      \[\leadsto x + \color{blue}{y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-7}:\\ \;\;\;\;z \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y - x\right)\\ \mathbf{if}\;z \leq -230:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- y x))))
   (if (<= z -230.0) t_0 (if (<= z 1.2e-9) (* (- 1.0 z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y - x);
	double tmp;
	if (z <= -230.0) {
		tmp = t_0;
	} else if (z <= 1.2e-9) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y - x)
    if (z <= (-230.0d0)) then
        tmp = t_0
    else if (z <= 1.2d-9) then
        tmp = (1.0d0 - z) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y - x);
	double tmp;
	if (z <= -230.0) {
		tmp = t_0;
	} else if (z <= 1.2e-9) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y - x)
	tmp = 0
	if z <= -230.0:
		tmp = t_0
	elif z <= 1.2e-9:
		tmp = (1.0 - z) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y - x))
	tmp = 0.0
	if (z <= -230.0)
		tmp = t_0;
	elseif (z <= 1.2e-9)
		tmp = Float64(Float64(1.0 - z) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y - x);
	tmp = 0.0;
	if (z <= -230.0)
		tmp = t_0;
	elseif (z <= 1.2e-9)
		tmp = (1.0 - z) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -230.0], t$95$0, If[LessEqual[z, 1.2e-9], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -230 or 1.2e-9 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

      3. lower--.f64 99.7

         \[\leadsto \color{blue}{\left(y - x\right)} \cdot z \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot z} \]

   if -230 < z < 1.2e-9

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(x \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{z \cdot x}\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot x} \]

      4. mul-1-neg N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \cdot x \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

      10. lower--.f64 78.7

          \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(1 - z\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -230:\\ \;\;\;\;z \cdot \left(y - x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) \cdot x\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-118}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 z) x)))
   (if (<= x -1.5e-53) t_0 (if (<= x 1.3e-118) (* z y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - z) * x;
	double tmp;
	if (x <= -1.5e-53) {
		tmp = t_0;
	} else if (x <= 1.3e-118) {
		tmp = z * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - z) * x
    if (x <= (-1.5d-53)) then
        tmp = t_0
    else if (x <= 1.3d-118) then
        tmp = z * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - z) * x;
	double tmp;
	if (x <= -1.5e-53) {
		tmp = t_0;
	} else if (x <= 1.3e-118) {
		tmp = z * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - z) * x
	tmp = 0
	if x <= -1.5e-53:
		tmp = t_0
	elif x <= 1.3e-118:
		tmp = z * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - z) * x)
	tmp = 0.0
	if (x <= -1.5e-53)
		tmp = t_0;
	elseif (x <= 1.3e-118)
		tmp = Float64(z * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - z) * x;
	tmp = 0.0;
	if (x <= -1.5e-53)
		tmp = t_0;
	elseif (x <= 1.3e-118)
		tmp = z * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.5e-53], t$95$0, If[LessEqual[x, 1.3e-118], N[(z * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5000000000000001e-53 or 1.3e-118 < x

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(x \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{z \cdot x}\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot x} \]

      4. mul-1-neg N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \cdot x \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

      10. lower--.f64 81.3

          \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\left(1 - z\right) \cdot x} \]

   if -1.5000000000000001e-53 < x < 1.3e-118

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 74.9

         \[\leadsto \color{blue}{y \cdot z} \]

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-118}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -230:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -230.0) (* z y) (if (<= z 8.2e-10) (* 1.0 x) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -230.0) {
		tmp = z * y;
	} else if (z <= 8.2e-10) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-230.0d0)) then
        tmp = z * y
    else if (z <= 8.2d-10) then
        tmp = 1.0d0 * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -230.0) {
		tmp = z * y;
	} else if (z <= 8.2e-10) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -230.0:
		tmp = z * y
	elif z <= 8.2e-10:
		tmp = 1.0 * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -230.0)
		tmp = Float64(z * y);
	elseif (z <= 8.2e-10)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -230.0)
		tmp = z * y;
	elseif (z <= 8.2e-10)
		tmp = 1.0 * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -230.0], N[(z * y), $MachinePrecision], If[LessEqual[z, 8.2e-10], N[(1.0 * x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -230 or 8.1999999999999996e-10 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 55.8

         \[\leadsto \color{blue}{y \cdot z} \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -230 < z < 8.1999999999999996e-10

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(x \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{z \cdot x}\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot x} \]

      4. mul-1-neg N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \cdot x \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

      10. lower--.f64 78.7

          \[\leadsto \color{blue}{\left(1 - z\right)} \cdot x \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(1 - z\right) \cdot x} \]

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 77.3%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -230:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ z \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z y))

C

double code(double x, double y, double z) {
	return z * y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * y
end function

Java

public static double code(double x, double y, double z) {
	return z * y;
}

Python

def code(x, y, z):
	return z * y

Julia

function code(x, y, z)
	return Float64(z * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * y;
end

Wolfram

code[x_, y_, z_] := N[(z * y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot z} \]
4. Step-by-step derivation

   1. lower-*.f64 41.5

      \[\leadsto \color{blue}{y \cdot z} \]

5. Applied rewrites 41.5%

   \[\leadsto \color{blue}{y \cdot z} \]

6. Final simplification 41.5%

   \[\leadsto z \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))