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

Percentage Accurate: 100.0% → 99.0%

Time: 5.5s

Alternatives: 7

Speedup: 1.2×

• 20.5% 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: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-lft-in N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 59.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{-137}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+77}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.9e-137)
   (* z y)
   (if (<= z 1.0) (* 1.0 x) (if (<= z 5.6e+77) (* (- z) x) (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.9e-137) {
		tmp = z * y;
	} else if (z <= 1.0) {
		tmp = 1.0 * x;
	} else if (z <= 5.6e+77) {
		tmp = -z * 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 <= (-6.9d-137)) then
        tmp = z * y
    else if (z <= 1.0d0) then
        tmp = 1.0d0 * x
    else if (z <= 5.6d+77) then
        tmp = -z * 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 <= -6.9e-137) {
		tmp = z * y;
	} else if (z <= 1.0) {
		tmp = 1.0 * x;
	} else if (z <= 5.6e+77) {
		tmp = -z * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.9e-137:
		tmp = z * y
	elif z <= 1.0:
		tmp = 1.0 * x
	elif z <= 5.6e+77:
		tmp = -z * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.9e-137)
		tmp = Float64(z * y);
	elseif (z <= 1.0)
		tmp = Float64(1.0 * x);
	elseif (z <= 5.6e+77)
		tmp = Float64(Float64(-z) * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.9e-137)
		tmp = z * y;
	elseif (z <= 1.0)
		tmp = 1.0 * x;
	elseif (z <= 5.6e+77)
		tmp = -z * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.9e-137], N[(z * y), $MachinePrecision], If[LessEqual[z, 1.0], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 5.6e+77], N[((-z) * x), $MachinePrecision], N[(z * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.89999999999999976e-137 or 5.60000000000000001e77 < z

   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. *-commutative N/A

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

      2. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -6.89999999999999976e-137 < z < 1

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.1%

      \[\leadsto 1 \cdot x \]

   if 1 < z < 5.60000000000000001e77

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 63.2

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{-137}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+77}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-143} \lor \neg \left(z \leq 0.00175\right):\\ \;\;\;\;z \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.5e-143) (not (<= z 0.00175))) (* z (- y x)) (* (- 1.0 z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.5e-143) || !(z <= 0.00175)) {
		tmp = z * (y - x);
	} else {
		tmp = (1.0 - z) * x;
	}
	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 <= (-3.5d-143)) .or. (.not. (z <= 0.00175d0))) then
        tmp = z * (y - x)
    else
        tmp = (1.0d0 - z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.5e-143) || !(z <= 0.00175)) {
		tmp = z * (y - x);
	} else {
		tmp = (1.0 - z) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.5e-143) or not (z <= 0.00175):
		tmp = z * (y - x)
	else:
		tmp = (1.0 - z) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.5e-143) || !(z <= 0.00175))
		tmp = Float64(z * Float64(y - x));
	else
		tmp = Float64(Float64(1.0 - z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.5e-143) || ~((z <= 0.00175)))
		tmp = z * (y - x);
	else
		tmp = (1.0 - z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.5e-143], N[Not[LessEqual[z, 0.00175]], $MachinePrecision]], N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000005e-143 or 0.00175000000000000004 < z

   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. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if -3.50000000000000005e-143 < z < 0.00175000000000000004

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-143} \lor \neg \left(z \leq 0.00175\right):\\ \;\;\;\;z \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-10} \lor \neg \left(x \leq 3.2 \cdot 10^{-208}\right):\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.3e-10) (not (<= x 3.2e-208))) (* (- 1.0 z) x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.3e-10) || !(x <= 3.2e-208)) {
		tmp = (1.0 - z) * 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 ((x <= (-1.3d-10)) .or. (.not. (x <= 3.2d-208))) then
        tmp = (1.0d0 - z) * 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 ((x <= -1.3e-10) || !(x <= 3.2e-208)) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.3e-10) or not (x <= 3.2e-208):
		tmp = (1.0 - z) * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.3e-10) || !(x <= 3.2e-208))
		tmp = Float64(Float64(1.0 - z) * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.3e-10) || ~((x <= 3.2e-208)))
		tmp = (1.0 - z) * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.3e-10], N[Not[LessEqual[x, 3.2e-208]], $MachinePrecision]], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.29999999999999991e-10 or 3.2000000000000001e-208 < x

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.29999999999999991e-10 < x < 3.2000000000000001e-208

   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. *-commutative N/A

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

      2. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-10} \lor \neg \left(x \leq 3.2 \cdot 10^{-208}\right):\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{-137} \lor \neg \left(z \leq 8.5 \cdot 10^{-8}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.9e-137) (not (<= z 8.5e-8))) (* z y) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.9e-137) || !(z <= 8.5e-8)) {
		tmp = z * y;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-6.9d-137)) .or. (.not. (z <= 8.5d-8))) then
        tmp = z * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.9e-137) || !(z <= 8.5e-8)) {
		tmp = z * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.9e-137) or not (z <= 8.5e-8):
		tmp = z * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.9e-137) || !(z <= 8.5e-8))
		tmp = Float64(z * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.9e-137) || ~((z <= 8.5e-8)))
		tmp = z * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.9e-137], N[Not[LessEqual[z, 8.5e-8]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.89999999999999976e-137 or 8.49999999999999935e-8 < z

   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. *-commutative N/A

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

      2. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -6.89999999999999976e-137 < z < 8.49999999999999935e-8

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{-137} \lor \neg \left(z \leq 8.5 \cdot 10^{-8}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 7: 42.5% 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 x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 45.7

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

5. Applied rewrites 45.7%

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

6. Final simplification 45.7%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))