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

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

• 20.2% 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, 0.1× 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 61.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-30}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+169}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= z -4.2e+158)
     t_0
     (if (<= z -3.6e-30)
       (* y z)
       (if (<= z 6.2e-65) x (if (<= z 7.5e+169) (* y z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (z <= -4.2e+158) {
		tmp = t_0;
	} else if (z <= -3.6e-30) {
		tmp = y * z;
	} else if (z <= 6.2e-65) {
		tmp = x;
	} else if (z <= 7.5e+169) {
		tmp = 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 = z * -x
    if (z <= (-4.2d+158)) then
        tmp = t_0
    else if (z <= (-3.6d-30)) then
        tmp = y * z
    else if (z <= 6.2d-65) then
        tmp = x
    else if (z <= 7.5d+169) then
        tmp = 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 = z * -x;
	double tmp;
	if (z <= -4.2e+158) {
		tmp = t_0;
	} else if (z <= -3.6e-30) {
		tmp = y * z;
	} else if (z <= 6.2e-65) {
		tmp = x;
	} else if (z <= 7.5e+169) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if z <= -4.2e+158:
		tmp = t_0
	elif z <= -3.6e-30:
		tmp = y * z
	elif z <= 6.2e-65:
		tmp = x
	elif z <= 7.5e+169:
		tmp = y * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (z <= -4.2e+158)
		tmp = t_0;
	elseif (z <= -3.6e-30)
		tmp = Float64(y * z);
	elseif (z <= 6.2e-65)
		tmp = x;
	elseif (z <= 7.5e+169)
		tmp = Float64(y * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (z <= -4.2e+158)
		tmp = t_0;
	elseif (z <= -3.6e-30)
		tmp = y * z;
	elseif (z <= 6.2e-65)
		tmp = x;
	elseif (z <= 7.5e+169)
		tmp = y * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[z, -4.2e+158], t$95$0, If[LessEqual[z, -3.6e-30], N[(y * z), $MachinePrecision], If[LessEqual[z, 6.2e-65], x, If[LessEqual[z, 7.5e+169], N[(y * z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.1999999999999998e158 or 7.49999999999999992e169 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. unsub-neg 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in z around inf 65.6%

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

      1. associate-*r* 65.6%

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

      2. mul-1-neg 65.6%

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

   7. Simplified 65.6%

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

   if -4.1999999999999998e158 < z < -3.6000000000000003e-30 or 6.20000000000000032e-65 < z < 7.49999999999999992e169

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in y around inf 61.0%

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

      1. *-commutative 61.0%

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

   4. Simplified 61.0%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

   if -3.6000000000000003e-30 < z < 6.20000000000000032e-65

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 80.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+158}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-30}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+169}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]

Alternative 3: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+57} \lor \neg \left(y \leq 2.6 \cdot 10^{+50}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.8e+57) (not (<= y 2.6e+50))) (* y z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.8e+57) || !(y <= 2.6e+50)) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - 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 <= (-3.8d+57)) .or. (.not. (y <= 2.6d+50))) then
        tmp = y * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.8e+57) || !(y <= 2.6e+50)) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.8e+57) or not (y <= 2.6e+50):
		tmp = y * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.8e+57) || !(y <= 2.6e+50))
		tmp = Float64(y * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.8e+57) || ~((y <= 2.6e+50)))
		tmp = y * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.8e+57], N[Not[LessEqual[y, 2.6e+50]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999999e57 or 2.6000000000000002e50 < y

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   5. Taylor expanded in x around 0 74.5%

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

      1. *-commutative 74.5%

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

   7. Simplified 74.5%

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

   if -3.7999999999999999e57 < y < 2.6000000000000002e50

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. unsub-neg 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+57} \lor \neg \left(y \leq 2.6 \cdot 10^{+50}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-32} \lor \neg \left(z \leq 8.4 \cdot 10^{-33}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.5e-32) (not (<= z 8.4e-33))) (* (- y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e-32) || !(z <= 8.4e-33)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - 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 ((z <= (-9.5d-32)) .or. (.not. (z <= 8.4d-33))) then
        tmp = (y - x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e-32) || !(z <= 8.4e-33)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.5e-32) or not (z <= 8.4e-33):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.5e-32) || !(z <= 8.4e-33))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.5e-32) || ~((z <= 8.4e-33)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.5e-32], N[Not[LessEqual[z, 8.4e-33]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999999e-32 or 8.4e-33 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-lft-in 97.6%

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

      6. associate-+r+ 97.6%

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

      7. *-commutative 97.6%

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

   5. Applied egg-rr 97.6%

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

   6. Taylor expanded in z around inf 98.4%

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

      1. distribute-lft-in 96.0%

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

      2. *-commutative 96.0%

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

      3. mul-1-neg 96.0%

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

      4. cancel-sign-sub-inv 96.0%

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

      5. *-commutative 96.0%

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

      6. distribute-lft-out-- 98.4%

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

   8. Simplified 98.4%

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

   if -9.4999999999999999e-32 < z < 8.4e-33

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. unsub-neg 79.3%

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

   4. Simplified 79.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-32} \lor \neg \left(z \leq 8.4 \cdot 10^{-33}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 5: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.9 \cdot 10^{-32}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.9e-32))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.9e-32)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (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 ((z <= (-1.0d0)) .or. (.not. (z <= 1.9d-32))) then
        tmp = (y - x) * z
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.9e-32)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.9e-32):
		tmp = (y - x) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.9e-32))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.9e-32)))
		tmp = (y - x) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.9e-32]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.90000000000000004e-32 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-lft-in 97.6%

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

      6. associate-+r+ 97.6%

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

      7. *-commutative 97.6%

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

   5. Applied egg-rr 97.6%

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

   6. Taylor expanded in z around inf 98.3%

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

      1. distribute-lft-in 95.9%

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

      2. *-commutative 95.9%

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

      3. mul-1-neg 95.9%

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

      4. cancel-sign-sub-inv 95.9%

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

      5. *-commutative 95.9%

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

      6. distribute-lft-out-- 98.3%

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

   8. Simplified 98.3%

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

   if -1 < z < 1.90000000000000004e-32

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in y around inf 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.9 \cdot 10^{-32}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-31} \lor \neg \left(z \leq 3.7 \cdot 10^{-65}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.7e-31) (not (<= z 3.7e-65))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.7e-31) || !(z <= 3.7e-65)) {
		tmp = y * z;
	} else {
		tmp = 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 <= (-1.7d-31)) .or. (.not. (z <= 3.7d-65))) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.7e-31) || !(z <= 3.7e-65)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.7e-31) or not (z <= 3.7e-65):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.7e-31) || !(z <= 3.7e-65))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.7e-31) || ~((z <= 3.7e-65)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.7e-31], N[Not[LessEqual[z, 3.7e-65]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.7000000000000001e-31 or 3.7e-65 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in y around inf 54.7%

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

      1. *-commutative 54.7%

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

   4. Simplified 54.7%

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

   5. Taylor expanded in x around 0 52.6%

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

      1. *-commutative 52.6%

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

   7. Simplified 52.6%

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

   if -1.7000000000000001e-31 < z < 3.7e-65

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 80.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-31} \lor \neg \left(z \leq 3.7 \cdot 10^{-65}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 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]

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]

2. Final simplification 100.0%

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

Alternative 8: 36.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]

2. Taylor expanded in z around 0 41.4%

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

3. Final simplification 41.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023326 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))