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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 6

Speedup: 1.2×

• 21.1% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.085:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y x) z)))
   (if (<= z -1.0) t_0 (if (<= z 0.085) (+ x (* y z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.085) {
		tmp = x + (y * z);
	} 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 = (y - x) * z
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 0.085d0) then
        tmp = x + (y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.085) {
		tmp = x + (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) * z
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 0.085:
		tmp = x + (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) * z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.085)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) * z;
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.085)
		tmp = x + (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 0.085], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.0850000000000000061 < 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. lower-*.f64 N/A

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

      2. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -1 < z < 0.0850000000000000061

   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 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.085:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-68}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y x) z)))
   (if (<= z -6.2e-52) t_0 (if (<= z 4.6e-68) (- x (* x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -6.2e-52) {
		tmp = t_0;
	} else if (z <= 4.6e-68) {
		tmp = x - (x * z);
	} 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 = (y - x) * z
    if (z <= (-6.2d-52)) then
        tmp = t_0
    else if (z <= 4.6d-68) then
        tmp = x - (x * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -6.2e-52) {
		tmp = t_0;
	} else if (z <= 4.6e-68) {
		tmp = x - (x * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) * z
	tmp = 0
	if z <= -6.2e-52:
		tmp = t_0
	elif z <= 4.6e-68:
		tmp = x - (x * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) * z)
	tmp = 0.0
	if (z <= -6.2e-52)
		tmp = t_0;
	elseif (z <= 4.6e-68)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) * z;
	tmp = 0.0;
	if (z <= -6.2e-52)
		tmp = t_0;
	elseif (z <= 4.6e-68)
		tmp = x - (x * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.2e-52], t$95$0, If[LessEqual[z, 4.6e-68], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999998e-52 or 4.59999999999999994e-68 < 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. lower-*.f64 N/A

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

      2. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -6.1999999999999998e-52 < z < 4.59999999999999994e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      4. *-rgt-identity N/A

         \[\leadsto \color{blue}{x} - x \cdot z \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{x - x \cdot z} \]

      6. *-commutative N/A

         \[\leadsto x - \color{blue}{z \cdot x} \]

      7. lower-*.f64 77.5

         \[\leadsto x - \color{blue}{z \cdot x} \]

   5. Applied rewrites 77.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-52}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-68}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-6}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+46}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e-6) (* y z) (if (<= y 1.9e+46) (* z (- x)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e-6) {
		tmp = y * z;
	} else if (y <= 1.9e+46) {
		tmp = z * -x;
	} else {
		tmp = y * z;
	}
	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 (y <= (-7d-6)) then
        tmp = y * z
    else if (y <= 1.9d+46) then
        tmp = z * -x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e-6) {
		tmp = y * z;
	} else if (y <= 1.9e+46) {
		tmp = z * -x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e-6:
		tmp = y * z
	elif y <= 1.9e+46:
		tmp = z * -x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e-6)
		tmp = Float64(y * z);
	elseif (y <= 1.9e+46)
		tmp = Float64(z * Float64(-x));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7e-6)
		tmp = y * z;
	elseif (y <= 1.9e+46)
		tmp = z * -x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e-6], N[(y * z), $MachinePrecision], If[LessEqual[y, 1.9e+46], N[(z * (-x)), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999989e-6 or 1.9e46 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if -6.99999999999999989e-6 < y < 1.9e46

   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. lower-*.f64 N/A

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

      2. lower--.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 39.9%

      \[\leadsto z \cdot \left(-x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (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 = (y - x) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lower-*.f64 N/A

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

   2. lower--.f64 65.7

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

5. Applied rewrites 65.7%

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

6. Final simplification 65.7%

   \[\leadsto \left(y - x\right) \cdot z \]
7. Add Preprocessing

Alternative 6: 42.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y * 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 = y * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 41.0

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

5. Applied rewrites 41.0%

   \[\leadsto \color{blue}{y \cdot z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))